Pre-Calc Final Exam

20 degrees 6’ 18” =

20.15

what does 6’ =

6 minutes

what does 18” =

18 seconds

standard position goes in what direction

counterclockwise

define coterminal angle

angle sin standard position that share the same terminal side

which of the following angles are coterminal with theta = 70 degrees

-70, 430, -290, -430, 790

430 -290, 790

what are the smallest positive angles that are coterminal with these angles?

490, -290, -70

70, 70, 290

define complementary angle

angles that add up to 90 degrees

define supplementary angles

angles that add up to 180 degrees

define radian

unit of angular measure when 2pi radians is equal to 360 degrees

convert the following angles to radians

180, 90, 45, 4, 60, 30

180 = pi

90 = pi/2

45 = pi/4

4 = pi/45

60 = pi/3

30 = pi/6

convert the following radians to angles

3pi, 3/2pi

3pi = 540

3/2pi = 270

are the angles 303 and -163 coterminal?

no

are the angles 2pi/3 and 14pi/3 in radians coterminal?

yes

what is the degree measure of the angle between 0 and 360 degrees that is coterminal with the -35 degree angle?

325 degrees

find the angle, in radian measure, between 0 and 2pi that is coterminal witht he given angle.

a) 13pi/5 is coterminal with:

b) -7pi/3 is coterminal with:

c) 77pi/2 is coterminal with:

d) 11pi/7 is coterminal with:

a) 3pi/5

b) 5pi/3

c) pi/2

d) 11pi/7

find the radian measure of each of the following angles in the interval [0,2pi).

a) -290 =

b) 40 =

c) -130 =

d) 400 =

e) -130 =

a) 4pi/9

b)2pi/9

c) 23pi/18

d) 2pi/9

e) 23pi/18

find the degree measure of each of the following angles given in radians in the interval [0,360)

a) -2pi/6

b) 9pi/4

c) -7pi/3

d) -5pi/2

e) 0pi

a) 300

b) 45

c) 300

d) 270

e) 0

determine which statement is true.

a) if the terminal side of an angle in standard position lies in quadrant I, then the angle is positive

b) if two angles in standard position have the same measure, then they are coterminal

c) if the initial and terminal sides of an angle coincide, then the measure of the angle is zero

d) if two positive angles are complementary, then both are acute

A, D

find the distance that the earth travels in one day in its path around the sun.

Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius of 93 million miles.

1600921 miles

if sin(a)=2/11 find the value these trigonometric functions:

C

A B

a) csc(a) =

b) cot(a) =

c) sec(b) =

d) tan(b) =

a) 11/2

b) 3sqrt(13)/2

c) 11/2

d) 3sqrt(13)/2

If sin(a)=2/11 find the value these trigonometric functions:

C

A B

a) cot(a) =

b) sin(a) =

c) tan(b) =

d) csc(b) =

a) 11/5

b) 5sqrt(146)/146

c) 11/5

d) sqrt(146)11

suppose that 0 < theta < pi/2 and cos(theta) = 8/17.

find that value of each of the trigonometric functions.

give the answer as an exact fraction.

a) sin(theta)

b) tan(theta)

c) sec(theta)

a) 11/`7

b) 15/8

c) 17/8

suppose that 0 < theta < pi/2 and sec(theta)=10/5.

find that value of each of the trigonometric functions.

give the answer as an exact fraction.

a) csc(theta)

b) sin(theta)

c) cos(theta)

d) tan(theta)

e) cot(theta)

a) 2/sqrt(3)

b) sqrt(3)/2

c) ½

d) sqrt(3)

e) 1/sqrt(3)

the captain of the ship at sea sights a lighthouse which is 260 ft tall.

the captain measures the angle of elevation to the top of the lighthouse to be 20 degrees.

how far is the ship from the base of the lighthouse rounded to the nearest foot?

714ft

a 36 ft ladder leans against a building so that the angle between the ground and the ladder is 76 degrees.

How high does the ladder reach on the building?

34.95 ft

define reference angle

the acute/smallest postiive angle formed between the terminal side of any angle and the x axis

determine which quadrant the terminal point is by the information given.

a) sin(t)>0 and cos(t)<0

b) sin(t)>0 and cos(t)>0

c) sin(t)<0 and cos(t)<0

d) sin(t)<0 and cos(t)>0

a) quadrant 2

b) quadrant 1

c) quadrant 3

d) quadrant 4

find which quadrant the terminal point is in and the reference angle.

a) theta = 10pi/3

b) theta = -pi/4

c) theta = -5pi/6

d) theta = 2

a) quadrant 3, reference angle = pi/3

b) quadrant 4, reference angle = pi/4

c) quadrant 3, reference angle = pi/6

d) quadrant 2, reference angle = pi - 2

find the reference angle in radians.

a) 11pi/5

b) -3/10

c) 7pi/5

a) pi/5

b) 3/10

c) 2pi/5

find the reference angle in degrees

a) 231

b) 345

c) -135

a) 51

b) 15

c) 45

find the value of the trigonometric function without finding the value of the angle

a) sin(a)=7/9 in quadrant II, cos(a) =

b) sin(a)=-1/3 in quadrant IV, cos(a) =

c) cos(a)=-5/9 in quadrant III, sin(a) =

d) cos(a)=4/11 in quadrant IV, sin(a) =

a) - sqrt(32)/9

b) sqrt(8)/3

c) - sqrt(56)/9

d) - sqrt(105)/11

determine the quadrant in which the point on the unit circle corresponding to each radian measure t lies.

a) t= 1/18 pi

b) t= 19/18 pi

c) t= 31/18 pi

d) t= 5/9 pi

a) 1

b) 3

c) 4

d) 2

find the value of each of the following cosines of given angles.

a) cos(pi/6)

b) cos(pi/4)

c) cos(pi/3)

d) cos(pi/2)

e) cos(pi)

f) cos(2pi)

a) sqrt(3)/2

b)1/sqrt(2) or sqrt(2)/2

c) ½

d) 0

e) -1

f) 1

evaluate the following trigonometric expressions.

a) sin(-3pi/2)

b) cos(-pi)

c) tan(pi/6)

d) cot(-pi/4)

e) sec(4pi/3)

f) csc(pi/3)

a) 1

b) -1

c) sqrt(pi/6)

d) -1

e) -2

f) 2/sqrt(3)

find the point on the unit circle corresponding to each of the following angles.

a) 8pi

b) -13pi/2

a) (1,0)

b) (0,-1)

if the terminal point is (3/5, -4/5), find the values of the following trigonometric functions

a) sin(t)

b) cos(t)

c) tan(t)

a) -4/5

b) 3/5

c) -4/3

if the point P = (-sqrt(2)/2, y) is on the unit circle in quadrant II, find the value of y.

y = sqrt(2)/2

arcsin is in which quadrants

1 and 4

arccosine is in which quadrants

1 and 2

arctan is in which quadrants

1 and 4

sinx=1/2, what is x in radians?

pi/6, -7pi/6

cosx=sqrt(2)/2, what is x in radians?

pi/4, 7pi/4

simplify arcsin(sin(-pi/6))

-pi/6

simplify arccos(cos(7pi/6))

5pi/6

evaluate the following inverse trigonometric expressions in radians with -pi/2 < theta < pi.

a) arcsin(1)

b) arcsin(-1/2)

c) arccos(-1/2)

d) arccs(-sqrt(2)/2)

a) pi/2

b) -pi/6

c) 2pi/3

d) 3pi/4

evaluate the following inverse trigonometric expressions in radians with -pi/2 < theta < pi/2.

a) arctan(-sqrt(3))

b) arctan(-1)

c) arctan(-sqrt(3)/3)

a) -pi/3

b) -pi/4

c) -pi/6

evaluate the following inverse trigonometric expressions in radians with -pi/2 < theta < pi/2.

a) arcsin(sin(2pi/3))

b) arcsin(sin(7pi/6))

c) arccos(cos(pi/6))

d) arccos(cos(-pi/3))

a) pi/3

b) -pi/6

c) pi/6

d) pi/3

evaluate the following expressions

a) cos(arcsin(1))

b) tan(arcsin(sqrt(2)/2))

a) 0

b) 1

f(x)=-5/6(sin(2/3x-3)+5

a) amplitude

b) period

c) phase shift

d) vertical translation

a) 5/6

b) 3pi

c) 9/2

d) 5

f(x)=-7/12cos(pi/5x+4)+6

a) amplitude

b) period

c) phase shift

d) vertical translation

a) 7/12

b) 10

c) -20/pi

d) 6

determine if each of the functions below is odd, even, or neither odd or even

a) cos(x)

b) sin(x)cos(x)

c) sin²(x)

d) sin(x)

e) sin(x)

f) sin(x) + cos(x)

a) even

b) odd

c) even

d) odd

e) neither

find the positive values of a and b for tan²t - sin²t = sin^at/cos^bt

a = 4, b = 2

use identities to simplify the following.

a) sec²-1

b) sin(x)tan(x)/cos(x)

c) sec(x)cos(x)

a) (tan(x))²

b) (tan(x))²

c) 1

find all solutions in the interval [0,2pi) for 2cosx-1=0

pi/3, 5pi/3

find all solutions in the interval [0,2pi) for 2 =-2sinx+1

7pi/6, 11pi/6

find all solutions in the interval [0,2pi) for 3cot²x-1=0

pi/3, 2pi/3, 4pi/3, 5pi/3

find all solutions in the interval [0,2pi) for 5=2-4cosx

arccos(-3/4), 2pi-arccos(-3/4)

find all solutions in the interval [0,2pi) for 2-2sin²x=1+cosx

0, 2pi/3, 4pi/3

find all solutions in the interval [0,2pi) for 2cos3x=1

pi/9, 5pi/9, 7pi/9, 11pi/9, 13pi/9, 17pi/9

find all solutions in the interval [0,2pi) for (cosx)(2sinx)+1=0

pi/2, 3pi/2, 7pi/6, 11pi/6

find all solutions in the interval [0,2pi) for 2cos²t-cost-1=0

0, 2pi/3, 4pi/3

find all solutions in the interval [0,2pi) for 4sinxcosx+2inx-2cosx-1=0

pi/6, 5pi/6, 2pi/3, 4pi/3

sin(x+pi/6)=Asin(x)+Bcos(x). find A and B.

A = sqrt(3)/2

B = 1/2

sin(x-pi/2)=Acosx. find A.

A = -1

cos3pi/7cos2pi/21 + sin3pi/7sin2pi/21 = cospi/A = B

A = 3

B = 1/2

cos(x+pi/6) + Sin(x-pi/6) =

0

use an identity to find the exact trig expression values.

a) sin(pi/12)

b) sin(5pi/12)

c) cos(3pi/8)

d) cos(pi/12)

use double or half angle identity to find the exact value of each trig expression below if cos(x) = -7/9 with pi < x < 3pi/2

a) cos(2t)

b) sin(2t)

c) cos(t/2)

d) sin(t/2)

A) 17/81

B) 56sqrt(2)/81

C) -1/3

d) 2sqrt(2)/3

find all solutions in the interval [0,2pi) for sin2t+sint+4cost=-2

2pi/3, 4pi/3

if cos(a)=-3/5 and sin(a)>0, what are cos2a and sin2a

cos2a = -7/25

sin2a = -24/25

use power-reducing identities to write the trig expression using only first powers

cos^4t/16

1/64 (1+2cos(2t)+(1+cos(4t)/2))

find the exact value of cos(pi/8)

cos(pi/8) = sqrt(2+sqrt(2))/2

a=6, <c=60, <a=55. find <b, b, and c. sine law.

C

A B

<b = 65

b=6.64

c=6.34

c=6, <c=120, <b=50. find <a, a, b. sine law.

C

A B

<a = 10

a = 1.2034

b=5.31

to find the distance across the river (AB), a distance BC=220 ft is laid off on one side of the river as shown. It is found that <B=115 and <C=27. What is the distance across the river? sine law.

B C

A

162.2 ft

b=8,<C=120, <A=40. find <B, a, c. sine law.

C

A B

<B = 20

a= 15.05

c= 20.26

a) if a=27, b=23, and <A=35 how many triangles are possible? sine law.

b) if a=13, b=15, and <A=52 how many triangles are possible? sine law.

c) if a=10.8, b=23, and <A=42 how many triangles are possible? sine law.

a)1

b) 2

c) 0

cosine law. <C=60, b=4, a=6. find c, <A, <B.

A

C B

c= 2sqrt(7)

<a=79.11

<b=40.89

a triangular parcel of land has sides of length 670, 180, and 624 ft. use cosine law to find <C, <A, and <B

<C=15.5

<A=83.1

<B=98.6

a steep mountain is inlined 74 degrees to the horizontal and rises 3400 ft above the surrounding plain. a stable car is to be installed from a point 930 ft from the base tot he top of the mountain. find the shortest length of cable needed. use cosine law.

3270

compute the sums and differences of the vectors

a) <5,0> + <2,-1>

b) <5,0> - <2,-1>

A) <7,-1>

B) <3,1>

find the magnitude and direction angle for each vector where the degree measure is in 0 < theta < 360.

A) < - 6, 5>

B) < -2, -2>

A) magnitude = sqrt(61), direction angle = 140.19

B0 magnitude = 2sqrt(2), 225

find the magnitude of the horizontal and vertical components for the vector with magnitude and direction angle given below.

v = 1530

theta = 101.5

vx = 305

vy = 1500.6

u = <3,-3> and v=<2,5>. solve.

a) u + v

b) u - v

c) v - u

d) 4u

e) -1/7v

f) 4u - 7v

a) <5,2>

b) <1,-8>

c) < -1,8>

d) <12, -12>

e) < -2/7, -5/7>

f) < -2, -47>

a donkey and a large dog are both hooked to tow-ropes connected to a wagon stuck in the mud in the middle of the country road.

the donkey pulls with a force of 150 pounds at the angle of 5 degrees with respect to the road.

the dog pulls with the force of 20 pounds at an angle of -38 degrees with the respect to the road.

a) what is the magnitude of the resultant force?

b) what is the direction, with respect to the road, of the resultant force?

A) 165.15 lbs

B) 0.27

an ultralight flies northeast at 60 mi/hr. it encounters a wind from the north blowing at 25 mi/hr.

A) what is the bearing of the ultralight? direction

b) what is the ground speed of the ultralight? magnitude

a) 25 deg

b) 45.9 mi/hr

<10,3> . <2,7>

41

u=< -6,-5> and v=<4,-1>. compute…

a) ||u||

b) ||v||

c) u x v

a) sqrt(610

b) sqrt(17)

c) -19

what is the radian measure of the angle between the vectors u=< -5,-2> and v=< - 5,8>

arccos(9/sqrt(2581))

find the projection of u = <1,0> on v=<5,-2> onto v=< -5,-5> as well as the component of u orthogonal to v.

a) projvu

b) orthogonal component

a) < -0.5, -0.5>

b) < -0.5, 0.5>

let a =<2,5> and b=(-4,-4>. find decomposition a= a|| + at with respect to vector b

a) A||

b) At

a) <3.5, 3.5>

b) < -1.5, 1.5>

linear substitute and eliminate:

y=-x+2

y=4x-8

(2,0)

nonlinear substitute

x²+y²=25

x-2y=5

(-7/2, -17/4)

(1,-2)

find 2 numbers whose sum is 16 and whose squares have the sum of 146

11 and 5

substitute. find x and y values.

-x+y=1

4x-3y=-1

x=2, y=3

elimination. find x and y.

5x+27=-7

7x+3y=-10

x=-1, y=-1

solve the system. find x and y.

x² - 20x + y² = -75

4x + 3y = 15

x=6, y=-3

find all solutions of the system equation. enter answers in ordered pairs.

x² + 4y² = 4

2x² - 2y = 13

none

find all solutions of the system in ordered pairs.

x² - 2y = -2

x² + 5y = 19

(2,3), (-2,3)