Academic Pre-Calculus

Chapter 4

Trigonometry

4.5 Graphs of Sine and Cosine Functions

4.6 Graphs of Other Trigonometric Functions

4.7 Inverse Trigonometric Functions

4.5 Day 1: Graphing Sine and Cosine Functions

Remember from the unit circle:

So, the graph of y = sinθ looks like: And, the graph of y = cosθ looks

like:

y= sinx y= cosx

Given: y = asinbx or y = acosbx

a = amplitude, which is the distance from the x-axis to the top of the graph (always

positive!)

-a: reflects x-axis

b = number of cycles in a period from 0 to 2π

-b: reflects y-axis

period = 2π

b

increments = period

(0, 1)

(1, 0)

(0, -1)

(-1, 0) (cosx, sinx)

Sketch the graph of each function for one period.

Determine the amplitude and period for the following trig functions.

1) y = 3sin4x 2) y = 5 cos2x

3) y = - 4sin(

) 4) y =

cosx

Name:____________________ 4.5 Day 1:Sine and Cosine

Transformations Homework Sketch the graph of each function for one period.

Determine the amplitude and period for the following trig graphs.

1. y = 2 sin 3x 2. y = -2 cos 2x

3. y = 5 cos 4x 4. y = -sin (1/2x )

5. y = cos 2x 6. y = -3 sin x

7. y = 1

4 sin 2x 8. y = -cos x

9. y = 4 cos (1/2) x 10. y = -2sin 2x

11. y = -3cos 2x 12. y = 1

2 sin 2x

Warm – Up

Describe the transformation:

1) 2

3 5y x

2) 2

4 6y x

Factor out the 3:

3) 3 9y x

4) 3 1y x

5) 1

32

y x

4.5 Day 2: Graphing Sine and Cosine Functions

Given: y = asinb(x±h) ± k or y = acosb(x±h) ± k

a = amplitude, which is the distance from the x-axis to the top of the graph (always

positive!)

-a: reflects x axis

b = number of cycles in a period from 0 to 2π

-b: reflects y axis

h: phase shift (+ goes left, - goes right)

k: vertical shift (+ goes up, - goes down)

period = 2π

b

increments = period

Sketch the graph of each function for one period.

Determine the amplitude, period, phase shift and vertical shift for the following trig

functions.

1) y = (

) 2) y = (

)

3) y = ( ) 4) y = (

)

Name:____________________ 4.5 Day 2:Sine and Cosine Transformations

Homework

Sketch the graph of each function for one period. Determine the amplitude, period, phase shift, and vertical shift for each.

1. y = 2 sin (2x + 2π) 2. y = cos 2x − 5

3. y = sin (x − π) − 1 4. y = 3 sin x − 1

5. y = 4 sin (1/2) x + 2 6. y = 3 sin (2x-π) - 1

7. y = 1

4 sin (2x+2π) 8. y = -cos x -1

9. y = 4 sin π

x3

+ 2 10. y = cos 2(x − π)

11. y = 3 sin π

2x2

+ 1 12. y = cos 2x – 1

Warm Up

Sketch the graph of the function for one period. Determine the amplitude, period, phase shift, and vertical shift for each.

(

)

Name:____________________________ Date:___________________ Worksheet: Review for sine/cosine transformations

1. y = 4sin2x

2. y = -cos2x

3. y = -4sin(1/2x)

4.

y=cos (2x - π)+3

5.

3sin 2

4y x

Warm Up

Sketch the graph of the function for one period. Determine the amplitude, period, phase shift, and vertical shift for each.

(

)

4.6 Day 1: Graphs of Secant and Cosecant

Remember from graphs of sin and cos:

Given: y = asinb(x±h) ± k or y = acosb(x±h) ± k

a = amplitude, which is the distance from the x-axis to the top of the graph (always

positive!)

Secant and Cosecant graphs do not have amplitude. Instead, it is a vertical stretch.

Given: y = asecb(x±h) ± k or y = acscb(x±h) ± k

a = vertical stretch (always positive!) b = number of cycles in a period from 0

to 2π

-a: reflects x axis -b: reflects y axis

h: phase shift (+ goes left, - goes right) k: vertical shift (+ goes up, - goes down)

period = 2π

b increments =

period

Steps to graph secant or cosecant:

1) If csc, graph as if it were sin or if sec, graph as cos, but graph as a dotted line.

2) Graph vertical asymptotes: where the graph of sin or cos crosses the x-axis.

3) Go to each max and min vertex and graph the reciprocal (flip the graph).

Graph & determine the vertical stretch, period, phase shift ,vertical shift and

asymptotes.

1) y= csc -π

2) y= -2sec (

) 2

3) y= sec ( -2π

) -

4) y=- csc ( π

)

Name: _____________________________________ Period: _____ Date: ____________

4.6 Day 1 Homework

GRAPHS OF SECANT AND COSECANT FUNCTIONS

I. Determine the period, phase shift, and vertical shift, if any, of each function.

1. 4sec3( )y x Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

2. 2csc2( )y x Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

3. sec4

y x

Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

4. 2

3csc3

y x

Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

5. 2sec2( ) 3y x Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

6. 2

6csc 3 23

y x

Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

7. 3

2sec 4 34

y x

Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

8. 5csc3 5y x Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

9. 5

csc 46

y x

Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

10. 3

2sec 2 12

y x

Vertical Stretch _________Period _________ Phase Shift ________ Vertical Shift ________

II. Graph each function over a one period interval. Make sure to label axes.

1. 3csc2y x

2. 5sec2y x

3. 4csc2y x

4. 2secy x

5. 3

3csc24

y x

Warm Up

Graph each function over a one period interval. Make sure to label axes.

32sec 2 1

2y x

4.6 Day 2: Graphing Tangent and Cotangent

Tan is undefined at

π

2,

π

2 and

π

2

Given: y=atanb(x±h) ± k or y=acotb(x±h) ± k

a = vertical stretch a≥ or vertical shrink a<

h: phase shift (+ goes left, - goes right)

k: vertical shift (+ goes up, - goes down)

period = π

b

Instead of increments, we now have asymptotes (must have at least 3 asymptotes).

To find asymptotes, set bx = where undefined and solve for x.

Graph and state asymptotes.

1) y= tan

(0, 1)

(1, 0)

(0, -1)

(-1, 0)

Cot is undefined at

-π and π

Graph and state asymptotes.

2) y= cot

3) y= cot2 ( -π

)

(0, 1)

(1, 0)

(0, -1)

(-1, 0)

4) y= tan ( π

) -2

5) y= cot ( π

) -

Name: ___________________________ Date: ____________ Period: _________

Section 4.6 Day 2: Graphing Tangent and Cotangent Functions Worksheet Day 2

Graph the following.

1. y = −2 tan (2x)

2. y = 4 cot (x) + 4

3. y = π

tan 2x 23

4. y = 2 cot 1 2π

x 32 3

5. y = 1

cot4x 54

6. y = π

4cot 3x 64

7.

1 1y tan x

2 2 4

8.

y 2tan 2x4

Warm Up

Graph the following.

y 2cot 2x 1

3

Name: __________________________________

Graphing All Trig Functions Review Worksheet (Sin, Cos, Tan, Csc, Sec, & Cot)

Sketch the graph of each function for one period.

Determine the amplitude or vertical stretch, period, phase shift, and vertical shift for each.

1) y = −sin x + 1

2) y = −3 cos 2x

3) y =

π4csc x

2

4) y = 2 tan 4x

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

5) y = 2cot(x-2

)

6) y =

π2sec x

4

7) y =

1 π2csc x 1

2 3

8) y = 2sec 2x π 4

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

9) y = 2 sin(2x+2

) + 3

10) y = 3 cos (x − π) + 2

11) y = 1 π

2tan x 32 2

12) y =

π2cot x 3

4

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift

Amplitude/Vertical Stretch Period Phase Shift Vertical Shift