Untitled Flashcards Set
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[Music]
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foreign
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hello and welcome to another geometry
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lesson by emath instruction my name is
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Kirk Weiler and today we'll be doing
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Unit 8 lesson three on trigonometry and
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the calculator so what we're going to be
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doing today is we're going to be getting
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used to more used to our three basic
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trigonometric ratios along with how we
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can use our calculator just a bit to
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work with trigonometry so make sure to
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have your calculators handy okay you
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want to have your sign your your
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graphing calculator and you want to have
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it in what's known as degree mode so
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make sure it thinks that you're working
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with angles in degrees otherwise you
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might get yourself into trouble anyway
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let's do a little review on what we've
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seen so far with trigonometry in a
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previous lesson we saw how similar right
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triangles gave rise to the three
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fundamental trigonometric ratios it is
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important to note the definitions of
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these three ratios now before we even
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move on right let's get that mnemonic up
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on the board right sohcahtoa
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the sine ratio whoops that that that was
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supposed to be a c and it didn't happen
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because I'm trying to talk and write at
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the same time
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all right the sine ratio is the opposite
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side divided by the hypotenuse cosine is
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adjacent divided by hypotenuse tangent
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is opposite divided by adjacent
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sometimes teachers will put in little
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fraction bars like this to help students
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remember that it's opposite divided by
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hypotenuse adjacent divided by
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hypotenuse opposite divided by adjacent
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right we've got to have that basic
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knowledge down because that is literally
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the way these ratios are defined for us
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let's take a look at exercise number one
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for the right triangle shown answer the
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following questions letter a state the
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value of each of the following ratios
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leave in fraction form okay so you know
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for instance if I want the sine of angle
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a I look at angle a I look at the leg
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that's opposite angle a and that's equal
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to 1 right I look at the hypotenuse
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that's equal to 2 and so the sine of
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angle a is simply equal to one-half
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right that is its ratio
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on the other hand the cosine of angle a
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right again here's angle a the leg
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adjacent to angle a is the square root
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of three the hypotenuse is equal to 2 so
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the cosine of angle a is equal to the
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square root of three
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divided by 2.
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and finally the tangent of angle a is
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the side opposite angle a divided by the
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leg adjacent to angle a so opposite of
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angle a we have one adjacent to angle a
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we have the square root of three and so
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the tangent of angle a is one divided by
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the square root of three
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some of your teachers might ask you to
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get rid of the radical in the
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denominator that's not an absolute
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requirement for me so I'm just going to
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leave it as 1 divided by the square root
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of 3 but you should obviously do
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whatever your teacher is asking you to
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do in terms of simplifying that fraction
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why don't you pause the video now and
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figure out what the values of sine C
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cosine C and tan CR
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all right so for the sine of angle C
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right I really focus on the angle the
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leg opposite of C is the square root of
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3 the hypotenuse Remains Two so the sine
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of angle C is root three over two
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right for the cosine of angle C the leg
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that's adjacent is equal to one again
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the hypotenuse is two so the cosine of
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angle C is one half
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and finally for the tangent of angle C
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we've got the side opposite of angle C
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which is root 3 the side adjacent which
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is equal to one so the tangent of C is
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root three over one which could be
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simplified to the square root of three
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but I'm going to just leave it like that
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all right great let's review something
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else we saw in the last lesson let's
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take a look at letter b
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what is the relationship between the
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sine and the cosine values of the
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complementary angles A and B and that
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actually should say the complementary
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angles A and C A and B are certainly not
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complementary to each other but a and C
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are right in fact for right now at least
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in this course we're not going to even
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talk about the sine the cosine or the
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tangent of 90 degree angles or angles
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larger than 90. we're just sticking with
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these two acute angles so what is the
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relationship between the sine and the
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cosine of the complementary angles A and
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C so don't worry about the tangent just
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look at these pause the video now and
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see if you can write something down
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all right well it's pretty easy right
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the sine of a is equal to the cosine of
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c and the cosine of a is equal to the
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sine of c and I'm going to write those
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down right here right so the sine of a
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is equal to the cosine of C
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INE of B
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and the sine of C is equal to the cosine
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of a
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so they are just switched
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all right and again that is always
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always true of two angles whose measures
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add up to 90. anytime two angles have
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measures that add up to the 90 the sine
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of one will equal the cosine of the
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other
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just like this
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all right so let's take a look
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at letter c
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using your calculator in degree mode
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which of the following is the correct
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value of the measurement of angle a use
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a guess and check approach we will see
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an easier way later
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okay now let's go back up to these
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values
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right
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don't worry about angle C I'm just
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trying to figure out how big angle a is
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I know the sine is one half the cosine
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is root 3 over 2 and the tangent is one
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divided by the square root of 3. the
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nicest of all of these is obviously sine
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of a equaling one half right so let's
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come back down to here
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we know the sine of a is equal to
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one-half so let's now take our
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calculator out and let's figure out
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let's ask it what is the sine of 20.
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what is the sine of 30.
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what is the sine of 40
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and what is the sine of 50 right as we
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discussed in our last lesson
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the calculator has these trigonometric
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ratio values stored in it for each angle
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both acute angles and not acute angles
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so all I have to do is bring my
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calculator up yet again let me remind
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you that there should be something that
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says degree right here or deg right here
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if you're using the ti Inspire if you're
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using let's say the ti 83 Plus or 84
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plus that's going to be located in a
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different location if you're using a
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calculus sorry any other type of
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calculator like a Casio or a Hewlett
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Packard or something like that you'll
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have to kind of look into how to change
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that mode but it's simple enough right I
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can hit my trig button
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I can hit enter on my sign
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and I can put in the sine of 20 and hit
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enter and the sine of 20 I'm going to
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just maybe round this to two decimal
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places 0.34
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so that's not right let's do the sine of
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30.
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and there's my winner 0.5
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right I don't even need to look at these
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okay and I don't need to look at them
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because
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the sine of 30 degrees is one half so
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angle a
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must measure 30 degrees
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all right that's got to be its
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measurement
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okay
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so the calculator again because of the
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similarity of right triangles that share
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acute angles the calculator has all of
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these ratios stored within it we're
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going to take advantage of that as we
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move ahead now since the values of the
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trigonometric trigonometric ratios only
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depend on the measure of a particular
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acute angle your calculator has all the
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values of these ratios stored in it
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let's take a look at exercise number two
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which of the following is closest to the
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length of NN in right triangle LMN shown
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below think about which trigonometric
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ratio makes the most sense given the
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angle and the one known side use your
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calculator to get and check and that
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should say guess and check I have all
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sorts of typos in this particular uh
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in this particular worksheet my
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apologies that should say guess and
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check now again we're going to see ways
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in future lessons to do this quicker but
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right now I only want to use the tools
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that are really available to us now what
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we're trying to figure out
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is the best choice the best choice not
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the exact choice but the best choice for
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that length
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now the one angle I know is my 42
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degrees
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that is opposite of 42
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is adjacent to 42. so what I need to use
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is my tangent ratio right my tangent
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ratio is opposite
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over adjacent right so it's all about
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tangent
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right the tangent of L
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will be opposite
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over adjacent so let's actually figure
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out let's just kind of take our
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calculator out
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and let's ask it what the tangent of 42
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degrees is
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simple enough
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okay
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so let's do trig
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we go over to tangent
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and we ask it for the tangent of 42
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degrees
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that's a pretty nice value that's 0.900
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four maybe I'll just stop it right there
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okay
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so right now keep in mind the tangent of
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that should be equal to NM divided by
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20. right so the tangent
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of L
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should be NM divided by 20. so the
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question is which one of those choices
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4nm will get us a ratio that is closest
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to
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0.9004 pause the video now and see if
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you can figure out which of these
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choices makes the most sense
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all right well let's just kind of go
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through it right we can do 14 divided by
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20.
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16 divided by 20.
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18 divided by 20.
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and 22 divided by 20. pretty sure it's
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not going to be the last one because
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that's going to be greater than 1 but
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right if I do 14 divided by 20 well of
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course 14 divided by 20.
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I'm going to get 0.7
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that doesn't look right 16 divided by
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20.
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and that's 0.8
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divided by 20.
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up I think that's it 0.9
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right
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0.9 0.9 roughly equal
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and so Choice 3 makes the most sense
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right now LM or NM is not going to be
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exactly equal to 18 but it's the closest
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based on the information that we're
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given in the problem
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all right let's do another one that's
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similar to this
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and hopefully has no typos in it you
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never know though exercise number three
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in right triangle ABC angle a is a right
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angle and the measure of angle C equals
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34 degrees
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which of the following measurements most
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closely match these conditions
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okay so this one's a little bit more
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tricky than the last one let's take a
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look at what we know so far we've got a
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right triangle we know a is the right
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angle and we know C measures 34 degrees
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so let's let's just kind of like get
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that drawn out
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okay so I've got my right triangle
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a is a right angle
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all right C
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and B
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C is 34 degrees and I don't really care
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what B is for right now
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okay
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now each one of these measurements will
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correspond to one particular Ratio or
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another for angle C
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but I think what makes sense is to right
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away just get some basic values for what
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the sine of 34 is the cosine of 34 and
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the tangent of 34. so I'm going to write
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those down here
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sine of 34 degrees
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cosine of 34 degrees
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tangent of 34 degrees what I'd like you
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to do right now
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is pause the video use your calculator
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and just figure out what each one of
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those is equal to to the nearest
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hundredth okay they're all going to be
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messy go ahead and do that
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all right let's do it
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here we go let me just clear this out a
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little bit oops that's not what I want
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try that again
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there it goes okay so first sine of 34
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. that is
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0.56 let's say rounded to the nearest
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hundredth let's now do cosine of 34. and
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that's going to be 0.83
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to the nearest hundredth and now let's
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get tangent
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of 34 and that's going to be
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0.67 to the nearest hundredth
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all right I'm going to put that away for
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a second
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all right so we know these
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one set of these values is going to
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correspond to one of these all right
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let's take a look at number one where we
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have a b equals 18.
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and we have BC is equal to 12.
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okay so right what would that allow me
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to do well the 18 is opposite C and the
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12 is the hypotenuse so that would be
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the sine ratio right this would be sine
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compared to angle C so in this case we'd
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have sine of C equals 18 divided by 12.
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and if I just do that real quickly on my
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calculator 18 divided by 12 I need that
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as a decimal that's 1.5
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all right
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since the sine of 34
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is not 1.5 that one is out all right
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that can't be the right choice
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all right simply the sine of C isn't 1.5
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okay let me just erase these
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all right what's my next one a b is
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equal to 14 and BC is equal to 25.
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that is again
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my sine ratio okay so my ratio stayed
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the same in this case right I still have
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a b and b c so in this case that would
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be the sine of C equals 14 divided by 25
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. let's go and see what that's equal to
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14 divided by 25
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and that's 0
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.56
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and there's my winner
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right sine of C 14 divided by 25 0.56
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sine of 34 is 0.56 that is my winner now
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if I worked through the last two which
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I'm not going to go through but if I did
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work through these two I'd find that
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none of them corresponded to these three
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values none of them would be correct
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none of them would give me the 0.83 or
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the 0.67
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all right let's keep working with our
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calculators and trigonometry now in the
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old days all right instead of having a
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calculator students had to read the
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trigonometric ratios from a table all
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right now before we kind of even get
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into this problem okay in a certain
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sense even though it's much easier to do
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trig on a calculator the tables were
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kind of nice because you could really
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see the ratios right
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um you know you'd have an angle like
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like in this particular table this is an
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everything right but you'd have an angle
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like 25 degrees and you'd be able to see
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the sine ratio the cosine ratio and the
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tangent ratio and having them kind of
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laid out like this gave you a sense for
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how those ratios kind of changed as the
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angle itself changed right notice like
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as the angle gets bigger your sine ratio
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gets bigger but your cosine ratio is
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actually getting smaller all right now
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again we won't get into that right now
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but you'll look at that a little bit
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more in algebra 2. let's take a look at
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exercise number four
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a partial table for the trigonometric
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ratios is shown below all ratios have
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been rounded to four decimal places
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answer the following questions based on
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the table
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letter A without using your calculator
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what must be the value of the cosine of
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63 degrees x explain
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alright so my claim is somehow without
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using your calculator you can figure out
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from this table what the cosine of 63
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degrees is
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see if you can figure out why that
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figure out why that is pause the video
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now
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well
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what should be true is that the cosine
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of 63
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should be equal to the sine of the
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complement of 63. all right so without
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using my calculator I think I can do
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this 90 minus 63 is equal to 27. all
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right so the cosine
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of 63 degrees should be the sign
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of 27 degrees
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all right and that I have
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that's right here
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so the cosine of 63 degrees should be
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0.4540
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and by the way that is the main reason
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why you're supposed to know that fact
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about the cosine of one angle being the
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sine of its complement in the old days
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these tables only went up to 45 degrees
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because you didn't need to go past 45
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given that you could always use this
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kind of relationship
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let's take a look at letter b
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which angle in the table is closest to
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the measure of angle a shown in the
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right triangle below justify your answer
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using a trigonometric ratio all right
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well how in the world am I going to use
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this table to figure out the measure of
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angle a
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well if I look at this particular right
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triangle right and I hone in on angle a
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it's always about the angle the 50
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the 50 is the leg that is opposite angle
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a
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and the 77 is the length of the leg
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adjacent to angle a so what I can say
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right is I can say that the tangent of
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angle a which is opposite divided by
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adjacent
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is equal to 50 divided by 77
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and let me figure out what that's equal
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to
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let me just get rid of all this mess
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50 divided by 77
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is
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0.64935
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Etc
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okay and now
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if I look at my table
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right and I go to my tangent column and
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I look for this particular value 0.649
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let's see do we have it ah there it is
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0.6494
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that looks like a is 33 degrees
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or maybe I should say the measure of
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angle a is 33 degrees
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all right
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that's it right we can use that table to
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help us figure out what the measure of
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angle a is but there's actually a
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quicker way to do it let's take a look
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at part C
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using your calculate a calculator
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evaluate each of the following round to
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the nearest whole number all right so
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now you got to watch out because when
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you look at each one of these there's
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your sign your cosine and your tangent
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but notice that there's a little
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negative one that looks like an exponent
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on each one of those maybe let's go to
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the screen for a minute so people can
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really see that right it says sine with
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a little negative 1 as an exponent
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cosine with a little negative 1 as an
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exponent and tangent with a little
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negative one as its exponent right those
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are what are called the inverse sign
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inverse cosine and inverse tangent so
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let's see how we get those on our
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calculator we actually see them exactly
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in the same place as the trig if I hit
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my trig button okay that's awesome
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um and I I go to sign with a little
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negative one
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and now I put in
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0.4384 and I hit enter
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rounded to the nearest whole number
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that's 26.
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right now if I do the cosine one whoops
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that's not what I wanted
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if I do the cosine one
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and I go down to the cosine with the
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little negative 1 called the inverse
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cosine
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and I put in 0.8290
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hit enter I'll get
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34
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and finally if I hit trig and I go over
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to the inverse tangent
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and I put in 0.6009
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right I'll get
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31.
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all right
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let's take a look at letter d
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what do you think the purpose of the
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inverse trigonometric functions are
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from letter c
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well
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take a look at your table
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right now if I look at the signed column
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okay
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maybe I have to go up a little bit more
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and I find
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0.4384 notice that corresponds to 26
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degrees
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if I go to the cosine column and I point
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find
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.8290.8290 it's got to be somewhere in
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there there it is 0.8290 and I go across
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that corresponds to an angle of 34. and
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if I go to the tangent and I look at
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.60009 or 0.6009 right that corresponds
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to an angle of 31 degrees
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so each one of these is kind of going in
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the opposite direction it's taking the
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ratios
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and it's giving us the angles that have
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those ratios and that makes sense right
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if our calculator has this table sitting
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inside of it somewhere right then it can
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take an angle and give us the ratio but
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it can also take us the rate take the
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ratio and give us the angle
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so what do you think the purpose of the
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inverse trigonometric functions are
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they find
24:35
the angle
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a particular
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trigonometric ratio trigonal
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no metric
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it's a long word
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trigonometric ratio
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and that will make finding angles in
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right triangles whose side lengths are
25:10
known exceptionally easy let's take a
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look at how that's done right the
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inverse trigonometric functions let's
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talk about these for a minute and then
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we'll see how to use them
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the inverse sine or sine with a little
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negative 1 x the inverse cosine the
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inverse tangent functions give an output
25:30
that is the angle that has the sine
25:32
cosine or tangent ratio that is the
25:36
input to the function all right
25:39
that seems weird let's see how it works
25:41
in practice exercise number five
25:45
given right triangle c d e shown below
25:48
letter A
25:49
what trigonometric ratio will relate
25:52
angle e to the given side lengths set up
25:55
the ratio all right so in other words
25:59
here I've got angle e
26:02
all right I know this side of 8 and I
26:04
know this side of 11.
26:06
what ratio can I write down based on
26:10
that angle at these two sides pause the
26:13
video now and see if you can figure that
26:15
out
26:20
all right here we go
26:24
to relate to angle e the eight is the
26:28
adjacent side
26:29
and the 11
26:31
is the hypotenuse
26:33
so of the three ratios it's the cosine
26:36
ratio that has the adjacent side and the
26:39
hypotenuse in it so I can say that the
26:41
cosine of angle e is equal to 8 divided
26:46
by 11.
26:49
okay great so I know what the cosine of
26:52
e is equal to it's equal to 8 11 right
26:55
which is some ugly decimal one or
26:58
another right now let's look at B
27:01
use the inverse trigonometric function
27:03
from part A to determine the measure of
27:05
angle e to the nearest degree this is
27:08
cool because I can now say since the
27:11
cosine of e is equal to 8 11 I can say
27:15
that the measure of angle e is equal to
27:18
the inverse cosine
27:20
of eight elevenths now you want to watch
27:23
out
27:24
it's very easy early on for students to
27:27
say well then the measure of angle e has
27:28
to be equal to the cosine of 8 11. no no
27:31
the cosine is eight elevenths what I
27:36
want to know is what angle has that
27:38
ratio but that's simple because my now I
27:42
can take my calculator out hit trig
27:45
go down to the inverse cosine
27:48
type in 8 divided by 11
27:52
hit enter
27:54
and the measure of angle E and A lot of
27:57
times students will just say E equals
27:59
they won't say the measure of angle e
28:01
that's okay
28:02
is 43 degrees
28:06
measured to the nearest
28:09
degree
28:11
all right how easy is that so so simple
28:16
right to figure out and of course it
28:18
would make some of the problems that we
28:19
did earlier in the lesson very very easy
28:21
right as long as we know two sides of
28:25
any right triangle we can use then the
28:27
inverse trig functions to come up with
28:30
either of the two acute angle measures
28:33
by just simply doing the inverse sine
28:35
the inverse cosine or the inverse
28:37
tangent and then putting the ratio in
28:39
there
28:40
all right let's summarize
28:43
and let me put my calculator away now so
28:46
today we reviewed what the three
28:49
trigonometric ratios were and how they
28:51
relate angle sizes to the three sides of
28:55
right triangles all right we also saw
28:57
how to bring the calculator into that by
29:00
doing some guess and check work and also
29:03
by using the inverse sign the inverse
29:05
cosine and the inverse tangent to figure
29:07
out the size of an angle the degree
29:09
measure of an angle if we knew the
29:12
particular sine cosine or tangent ratio
29:15
all right and that's going to be one of
29:17
the two main things that we do with
29:18
trigonometry we're going to really just
29:20
use trig to do one of two things either
29:22
find the size of an angle in a right
29:24
triangle or if we know the angle and one
29:28
of the sides to find one of the other
29:30
sides and we're going to concentrate
29:31
that on that a lot more in the next
29:33
lesson for now I just want to thank you
29:35
for joining me for another geometry
29:37
lesson by emath instruction my name is
29:40
Kirk Weiler and until I see you again
29:42
keep thinking
29:44
and keep solving problems