Precalc Unit 4 Test

sections 4.1-4.7

complete revolution or circle

360 degrees or 2 pi radians

pi in standard position

180 degrees

pi/2 in standard position

90 degrees

pi/3 in standard position

60 degrees

pi/6 in standard position

30 degrees

clockwise rotation from x-axis

negative angle

anti-clockwise rotation from x-axis

positive angle

initial ray

lies along the positive x-axis and serves as the reference line for measuring angles in standard position.

terminal ray

the ray that defines the angle's endpoint after rotation from the initial ray.

angle lies in quad 1

terminal side lies in quad 1

angle lies in quad 2

terminal side lies in quad 2

angle lies in quad 3

terminal side lies in quad 3

angle lies in quad 4

terminal side lies in quad 4

angle is a quadrantal angle

if the terminal side lies on the x-axis or y-axis and the angle measures 0°, 90°, 180°, 270°, or 360°.

coterminal angles

two angles with the same initial and terminal sides

find a positive angle less than 360 degrees that is coterminal with 400 degrees

400-360 = 40 degrees is the coterminal angle since it lies in the first quadrant.

find a positive angle less than 360 degrees that is coterminal with -135 degrees

-135 + 360 = 225 degrees is the coterminal angle since it lies in the third quadrant.

find a positive angle less than 360 degrees that is coterminal with 855 degrees

855 - 720 = 855 - (360(2))=135 degrees is the coterminal angle since it lies in the first quadrant.

linear speed

speed a particle moves along an arc of the circle (v) (distance is s and t is time). It is calculated as v = s/t, where v is measured in units per time.

angular speed

speed which the angle is changing as a particle moves along an arc of the circle. (angle measures in radians and t is time). w=angle/t

unit circle

a circle of radius 1 with its center at the origin and a rectangular coordinate system. the equation is x²+y²=1

trigonometric functions

circular functions; cosine (cos), sine (sin), tangent (tan), cotangent (cot), cosecant (csc), secant (sec)

sin t =

y

csc t

1/y or 1/sin t

sec t

1/x or 1/cos t

cot t

x/y or cos t/ sin t

cos t =

x

tan t

sin t/ cos t or y/x

trig functions when t = pi/4

a²+b²=c²; turns into a²+a²=1; since a = b and c=1 since radius of unit circle is 1

reciprocal IDs (csc t)

1/sin t

reciprocal IDs (sec t)

1/cos t

reciprocal IDs (cot t)

1/tan t

reciprocal IDs (sin t)

1/csc t

reciprocal IDs (cos t)

1/sec t

reciprocal IDs (tan t)

1/cot t

Quotient IDs (tan t)

sin t/cos t

Quotient IDs (cot t)

cos t/sin t

periodic functions with a period of 2pi

sin, cos, csc, sec; example..sin(t+2pi n)

periodic functions with a period of pi

tan and cot; example…tan(t + pi n)

right triangle definitions of Trig functions

sin=opp/hyp, cos=adj/hyp, and tan=opp/adj (soh cah toa)

sin 0 degrees = tan 0 degrees

=0

sin 30 degrees

=1/2

sin 45 degrees = cos 45 degrees

=1/ √2

sin 60 degrees = cos 30 degrees

=(squ rt 3)/2

sin 90 degrees = cos 0 degrees

=1

tan 30 degrees = cot 60 degrees

= 1/ (squ rt 3)

cos 90 degrees = cot 90 degrees

=0

tan 45 degrees = cot 45 degrees

=1

csc 45 degrees = sec 45 degrees

√2

sec 0 degrees and cot 0 degrees

undefined

tan 90 degrees and sec 90 degrees

undefined

csc 30 degrees = sec 60 degrees

=2

tan 60 degrees = cot 30 degrees

=squ rt 3

definitions of trig functions for any angle θ in standard position

• sinθ=y/r

• cosθ=x/r

• tanθ=y/x

• use reciprocal IDs to find cot, sec, and csc

signs of a trig function

if the angle is not quadrantal, then the sign of a trig function depends on the quadrant in which the terminal side of the angle lies.

Quad 1

x and y are positive so sin and cos are positive so all other trig functions are positive in quad 1

Quad 2

• x is negative and y is positive so cos and reciprocal sec are negative and sin is positive along with reciprocal csc

• both tan and cot are negative since the quotient of 2 opposite signs is negative

Quad 3

• x and y are negative so cos and sin are negative

• BUT tan and cot are positive because the quotient of two negatives is a positive

Quad 4

• x is positive so cos is positive along with sec, and y is negative so sin is negative along with csc

• tan and cot are negative since the quotient of two opposite signs is negative

ASTC for positive functions

1. A = all in quad 1

2. S = sin and reciprocal in quad 2

3. T = tan and reciprocal in quad 3

4. C = cos and reciprocal in quad 4

reference angle

an acute angle formed by the terminal side of a non-acute angle and the x-axis.

what is the reference angle of angle 345 degrees?

360 degrees (full unit circle) - 345 degrees = 15 degrees or the reference angle

what is the reference angle of the angle 5pi/6?

6pi/6 (or pi) - 5pi/6 = pi/6 as the reference angle

graphing y = sin x

• x= angle measures (in radians) and y= sin x so the graph is a curve

• Range: [-1,1]

• Domain: (all reals)

• Symmetry: odd function that is symmetrical about the origin

graphing y = cos x

• x= angle measures (in radians) and y=cos x so the graph is a curve

• Range: [-1,1]

• Domain (all reals)

• Symmetry: even function and symmetrical about the y-axis

comparison of y=cos x and y=sin x

• Range and Domain: SAME

• Period: SAME; 2 pi

• Intercepts: DIFFERENT

y = sin x intercepts

crosses thru origin and intercepts the x-axis at all multiples of pi (…-3pi, -2pi, -pi, 0, pi, 2pi, 3pi)

y = cos x intercepts

intercepts y-axis at (0,1) and intercepts x-axis at all odd multiples of pi/2 (…-3pi/2, -pi/2, pi/2, 3pi/2)

pi

3.14

2pi

6.28

pi/2

1.57

3pi/2

4.71

Amplitude

the greatest distance the curves rise and fall from the axis

• always positive since it represents a distance

Period

distance around the unit circle

Changing amplitude

if function value (y) is multiplied by a constant, THAT is the new amplitude

• example: y = sin x; amplitude is 1 but if y = 3sin x; amplitude is 3

changing the period

the length of the period is a function of the x-value

• example: y = sin x; period is 2pi but if y = sin(3x); period is 2pi/3

Period is 2pi divided by the coefficient of x

Phase Shift

graph is shifted left or right of the original graph

• example: y=sin(x- pi/3) so the graph is shifted right pi/3 units.

• change is made to the x-values, so it is addition or subtraction to the x-values

  • Subtraction shifts right

  • addition shifts left

Vertical Shift

graph can be shifted up or down on the coordinate axis by adding to the y-value

• example: y=sin x +3 so the graph of sin x moves up 3 units

y= Acos(Bx-C)+D

• amplitude is absolute value of A

• period: 2pi/B

• Phase Shift: C/B

• Vertical Shift: D

graphing y= tan x

NOTE: tan x =sin x/ cos x

• where cos x = 0 the tangent is undefined so at x= …-3pi/2, -pi/2, pi/2, 3pi/2…. the graph has vertical asymptotes

• x-intercepts where sin x = 0, x= ….-2pi, -pi, 0, pi, 2pi….

characteristics of y= tan x

• Period: pi

• Vertical asy.: odd multiples of pi/2

• Domain: (all reals except odd multiples of pi/2)

• Range: (all reals)

• odd function

• symmetric about origin

• x-intercepts: all multiples of pi (midway between the asymptotes)

• increasing function over (-pi/2, pi/2)

graphing y=cot x

NOTE: cot x= cos x/ sin x

• vertical asy: where sin x=0, (multiples of pi)

• x-intercepts: where cps x=0, (odd multiples of pi/2)

• decreasing function over (0,pi)

graphing y= csc x

• reciprocal of sin x

• vertical asy: where sin x=0, (x=integer multiples of pi)

• range: (∞-, -1]U[1,∞)

• Domain: all reals except integral multiples of pi

• period: 2pi

• odd function

• symmetry about origin

graphing y= sec x

• reciprocal of y= cos x

• vertical asy: where cos x=0 (odd multiples of pi/2)

• Range: (-∞, -1]U[1,∞)

• Domain: all reals except odd multiples of pi/2

• Period: 2pi

• even function

• symmetry about y-axis

What is the inverse sin of x?

The angle or real number that has a sin value of x

• example: the inverse sin of ½ is pi/6 (arcsin1/2=pi/6)

What is the shorthand notation for inverse sin x?

arcsin x or sin^-1x

Finding the Domain of y=sin^-1x

• the domain of any function becomes the range of its inverse

• the range of a function becomes the domain of its inverse

Range of y=sin x is [-1,1] so the domain of y= sin^-1x is [-1,1]

Trig values for special angles

if you know sin(pi/2)=1 then you know sin^-1(1)=pi/2

• Once you know the trig values for special angles, you also know the inverse trigs!!!

order of trig values in quad 1

• 0

• pi/6

• pi/4

• pi/3

• pi/2

order of trig values in quad 2

• pi/2

• 2pi/3

• 3pi/4

• 5pi/6

• pi

order of trig values quad 3

• pi

• -5pi/6

• -3pi/4

• -2pi/3

• -pi/2

order of trig values quad 4

• -pi/2

• -pi/3

• -pi/4

• -pi/6

• 0

inverse cosine function

• refers to the angle or real number that has a cosine of x

• arccos x or cos^-1x

• Domain: [-1,1]

• Range: [0,pi]

• quad 1 and 2

the inverse tangent function

• refers to the angle or real number that has a tangent of x

• arctan x or tan^-1x

• Domain: (all reals)

• Range: (-pi/2,pi/2)

• quad 1 and 4

evaluating compositions of functions and their inverses

what the function does, its inverse undoes

Find the exact value of cos(tan^-1(5/12))

cos(tan^-1(5/12)= 12/13

cos(cos^-1(0.6))=

0.6

cos(cos^-1(1.5))=

undefined because 1.5 is not in the domain [-1,1] and there is no angle that has a cosine greater than 1

find the exact value of sin^-1√2/2

pi/2