Honors Algebra 2 Chapter 14 Page 1

Section 14.1 (Introduction) – Graphs of Trig Functions Objectives: 1. To graph basic trig functions using t-bar method.

A. Sine and Cosecant

1. siny θ= 2. cscy θ=

B. Cosine and Secant 1. cosy x= 2. secy θ=

� y � y � y � y 0 --- --- 180 --- ---

30 120 210 300 45 135 225 315 60 150 240 330 90 --- --- 270 360

x y x y x y x y

0 --- --- π --- ---

6π 2 3π 4 3π 5 3π

4π 3 4π 5 4π 7 4π

3π 5 6π 7 6π 11 6π

2π --- --- 3 2π 2π

−360 −315 −270 −225 −180 −135 −90 −45 45 90 135 180 225 270 315 360

−1

1

−360 −315 −270 −225 −180 −135 −90 −45 45 90 135 180 225 270 315 360

−6

−4

−2

2

4

6

−3π/2 −π −π/2 π/2 π 3π/2 2π

−1

1

−3π/2 −π −π/2 π/2 π 3π/2 2π

−4

4

Honors Algebra 2 Chapter 14 Page 2

C. Tangent and Cotangent

1. tany x=

D. coty x=

−3π/2 −π −π/2 π/2 π 3π/2 2π

−4

4

−3π/2 −π −π/2 π/2 π 3π/2 2π

−4

4

x y x y x y x y

0 --- --- π --- ---

6π 2 3π 4 3π 5 3π

4π 3 4π 5 4π 7 4π

3π 5 6π 7 6π 11 6π

2π --- --- 3 2π 2π

Honors Algebra 2 Chapter 14 Page 3

Section 14.1 – Graphing Trigonometric Functions Objectives: 1. To find the amplitude and period for a trigonometric function. 2. To write equations of trigonometric functions given amplitude

and period.

I. Graphing Equations A. Forms:

1. 2.

B. What does A affect? 1. Graph:

a) 3siny θ= b) 5cosy θ= −

2. Amplitude ( )A – Distance from _____________________.

3. Example: a) Graph manually: 2.5siny θ= b) What is the amplitude of 7 cosy x= − ?

4. How does A− affect the graph? a) Graph: 3cosy x= − and 3cosy x= b) Graph: 3siny x= − and 3siny x= c) Graph manually: 7 cosy x= −

−360 −180 180 360

−4

−2

2

4

−2π −π π 2π

−8

−6

−4

−2

2

4

6

8

Honors Algebra 2 Chapter 14 Page 4

C. What does B affect? 1. Graph:

a) sin 3y x= b) cos( 2 )y x= −

2. ( )B -- ____________ of _____________ in 360° or 2π .

3. Period – ____________ to complete ______ cycle

or period� �=� �� �

(Note: tangent is in 180° or π .) 4. Examples:

a) Graph: 1

cos2

y x= (Note: When graphing, break your cycle into 4 equal parts)

b) Graph: tan 2y θ=

5. How does B− affect the graph? a) Graph: cos( 3 )y x= − and cos3y x= b) Graph: sin( 3 )y x= − and sin 3y x=

6. Graph manually: 1

cos4

y θ� �= −� �� �

Honors Algebra 2 Chapter 14 Page 5

D. Doing Both Amplitude and Period 1. Graph manually: 5cos3y x=

2. Graph manually: 1

2sin2

y x= −

3. Graph manually: 1

2sin3

y θ� �= −� �� �

II. Writing Equations: Examples:

1. Amplitude = 2 and Period = 90� 2. The motion of a prong of a tuning fork can be described by a modified trig function. Represent displacement of a prong from the rest positions to the right as positive and displacement to the left as negative. At time zero, a prong reaches its maximum displacement, 0.03-cm to the right. The fork has a frequency of 200 cycles per second.

a) Write a function that represents the position of the prong, y, in reference to its rest position, in terms of the time, t, measured from t = 0 seconds. b) Find y, when t = 0.02 seconds

Homework: p.766 – 1- 3 all, 15-33 odds, 35-37 all, 39-41 all, 43-50 all

Honors Algebra 2 Chapter 14 Page 6

Section 14.2– Translations of Trigonometric Graphs Objectives: 1. To find the phase shift and translations for a trigonometric function. 2. To graph trig functions with phase shifts and translations 3. To write equations of trigonometric functions given phase shifts and translations.

I. Graphing Equations A. Forms:

1. 2.

B. What does D affect? 1. Graph:

a) sin( 60 )y θ= − ° b) cos( )y x π= +

2. Phase Shift (D) – __________ the vertical___________ axis ________ or _________. 3. How does -D affect the graph?

Graph: cos4

y xπ� �= −� �

� � and cos

4y x

π� �= +� �� �

a) b)

4. Examples: a) Graph manually: sin( 45 )y θ= − ° b) What is the phase shift of cos( 180 )y θ= + ° ?

C. What does C affect? 1. Graph:

a) 3 siny x= + b) 2 cosy x= − +

2. Translation (C) – translating ________________ ______ or ________. 3. How does -C affect the graph? Graph: 3 cosy x= − + and 3 cosy x= +

a) b)

-630 -540 -450 -360 -270 -180 -90 90 180 270 360 450 540 630 720

-4

-2

2

4

Honors Algebra 2 Chapter 14 Page 7

4. Examples: a) Graph manually: 2 cosy x= − + b) What is the translation of 5 siny θ= − ?

D. Doing Both Phase Shift and Translation 1. Graph manually: 5 cos( )y x π= + − 2. Graph manually: 2 sin( 15 )y θ= − + + °

E. Basic Type of Transformations Review 1. Original Graph: sin ( )y C A B x D= + −

Rewrite using r

2. Horizontal shift r units to the right: ____________________

3. Horizontal shift r units to the left: ____________________

4. Vertical shift r units to the down: ____________________

5. Vertical shift r units to the up: ____________________

6. Reflection (about the x-axis): ____________________

7. Reflection (about the y-axis): ____________________

F. Doing it all

1. Graph manually: 1

5 3cos ( )2

y x π= − −

2. Graph manually: 1

2 6sin 154

y θ� �= − + − + °� �� �

-630 -540 -450 -360 -270 -180 -90 90 180 270 360 450 540 630 720

-4

-2

2

4

-630 -540 -450 -360 -270 -180 -90 90 180 270 360 450 540 630 720

-4

-2

2

4

Honors Algebra 2 Chapter 14 Page 8

3. Graph manually: 1

2 6csc 154

y θ� �= − + − + °� �� �

4. Graph manually: 1

5 3tan ( )2

y x π= − −

II. Writing Equations: Examples:

1. Amplitude = 2, Period = 90�, phase shift = 45� and translation = 4 2. As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. A particular wheel has a diameter of 38-ft., travels at a rate of 4 revolutions per minute, and the seats of the Ferris wheel clear the ground by 3-ft.

a) Write an equation to describe the change in height, h, of the seat that was filled last before the ride began in terms of time, t, in seconds. b) Find the height of the seat after 22-s.

Homework: worksheets and p. 775 – 44-46 all

Honors Algebra 2 Chapter 14 Page 9

Honors Algebra 2 Chapter 14 Worksheet Graphing Sinusoidal Functions in Radian Mode

Find the amplitude, period, phase (horizontal) displacement and translation (vertical displacement). Then use the information to find the critical points and sketch two cycles for the graph (one to the right and one to the left of the center point).

1. 1

3 2cos ( )5

y x π= + −

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

2. 2

4 5sin3 2

y xπ� �= − + +� �

� �

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

3. 2 6cos36

y xπ� �= − +� �

� �

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

4. 2 6sin ( 1)4

xπ+ −

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

5. 5 4sin ( 2)3

y xπ= − − +

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

Honors Algebra 2 Chapter 14 Page 10

Honors Algebra 2 Chapter 14 Worksheet Graphing Sinusoidal Functions in Degree Mode

Find the amplitude, period, phase (horizontal) displacement and translation (vertical displacement). Then use the information to find the critical points and sketch two cycles for the graph (one to the right and one to the left of the center point). 6. 7 4cos3( 10 )y θ= + − °

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

7. 10 20sin 2( 30 )y θ= − + + °

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

8. 1

3 5cos ( 90 )2

y θ= − + °

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

9. 1

1000 3000sin ( 60 )3

θ+ + °

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

10. 11 6sin( 17 )y θ= − − °

Amplitude: _________

Period: _________

Phase: _________

Translation: _________

Honors Algebra 2 Chapter 14 Page 11

Advanced Algebra 2 Chapter 14 Worksheet Graphing Tangent and Reciprocal Functions

Graph the following: Assume the function is circular (use radians) if the independent variable is x and trigonometric (use degrees) if the independent variable is θ. Graph two cycles of each (one to the right and left of the center point). 1. tan 2y θ=

2. cot3

y xπ=

3. csc2

y xπ=

4. sec3y θ=

5. 2coty x=

6. 1

tan2

y θ=

7. 1

sec3

y θ=

8. 2cscy θ=

9. 4 3tany xπ= +

10. 5 3cot 4y θ= − +

11. 6 2csc5y θ= − +

12. 1 4sec10

y xπ= +

13. 1 3cot 2( 30 )y θ= − + − °

14. 2 5 tan ( 3)8

y xπ= + −

15. 4 6sec ( 1)2

y xπ= + +

16. 3 2csc 4( 10 )θ− + + °

Honors Algebra 2 Chapter 14 Page 2

Section 14.3 – Basic Trig Identities

Objectives: 1. To identify and use reciprocal identities, quotient identities, and Pythagorean identities

I. Reciprocal Identities

A. Identities

1. 1 1

sin or csc or __________ 1

A A= = =

Note: 11sin

sinA

A−≠

2. 1 1

cos or sec or ___________ 1

A A= = =

3. 1 1

tan or cot or ___________ 1

A A= = =

B. Example: If 4

sec3

x = , find cos x .

II. Quotient Identities Identities

1.

tan A = Proof:

2.

cot A =

III. Pythagorean Identities

A. Identities 1. Proof: 2. 3.

B. Examples:

1. If 3

cos5

A = , find csc A .

2. Simplify: csc cos cotx x x−

3. Simplify the gravitational force formula: 2sec

sec tanmg x mg

Fx x

−=

Homework: p.779 – 13-35 odds, 37-44 all, 46-52 all

Honors Algebra 2 Chapter 14 Page 3

Section 14.4 – Verifying Trig Identities Objectives: 1. To use the basic trig identities to verify other identities. 2. To find the numeric values of trig functions

A. Method 1. _________: Transform the more ______________ side of the _____________ into the _________ of the ___________ side. 2. ____________: _______________ one or more basic trig ___________ to simplify the expression. 3. ____________: _________ or ____________ to simplify the expression. 4. Always repeat the ___________ and ___________ step until goal is met. 5. Tricks: Multiply both ____________ and _______________ by the ____________. Note: When trying to verify possible trig identities, you can only work with one side. If you perform an operation with both sides, then you are assuming they are equal

B. Examples

1. 2

tan sin costan

sinx x x

xx

− ≡

2. 1 tan

sin cossec

xx x

x+ ≡ +

3. Find the numeric value of one trig function: cot 3csc 5

xx

=

Homework: p.784 – 11-27 odds, 31, 32, 35, 36, 43-49 all

Honors Algebra 2 Chapter 14 Page 4

Section 14.5 – Sum and Difference Identities Objectives: 1. To use the sum and difference identities to find exact values and prove identities.

A. Identities 1. cos( )A B± = 2. sin( )A B± =

3.

tan( )A B± =

B. Examples

1. Find the exact value for cos 75° 2. Prove: cos( 180 ) cosA A+ ° ≡ − 3. Find the exact value for sin175° Hint: Find the exact value for sin15° 4. Prove: tan tan( )x xπ≡ +

5. If 5

sin13

A = such that 02

Aπ≤ ≤ and

35cos

37B

−= such that 32

Bππ ≤ ≤ , find

cos( )A B+ . Homework: p.788 – 15-39 odds, 40, 41, 48-60 all

Honors Algebra 2 Chapter 14 Page 5

Section 14.6 – Double Angle and Half-Angle Identities Objectives: 1. To use the double angle and half-angle identities to find exact values and prove identities.

A. Identities 1. Double Angle Identities

a) sin 2A = Proof:

b) cos 2A =___________________ or

___________________ or___________________

���

c)

tan 2A =

2. Half-Angle Identities

a)

sin2 2A = ± Proof:

b)

cos2 2A = ±

c)

tan , cos 12A

A= ± ≠ −

B. Examples

1. If 3

sin4

x = − and x terminates in quadrant IV. Find the exact values for:

a) sin 2x

b) 1

cos2

x

2. Prove: 2 21 cos 2 sec tanx x x− ≡ 3. Use the half-angle identity to find the exact value for sin15°

Homework: p.794 – 13-37 odds, 48-57 all

Honors Algebra 2 Chapter 14 Page 6

Section 14.7 – Solving Trig Equations Objectives: 1. To solve trig equations.

A. Examples 1. Solve 22sin sin 1 0x x+ − = for the principle values of x. 2. Solve sin 2 2x = − for all values of x. 3. Solve 4cos sin 2sin 2 3 cos 3 0x x x x− − + = for all values of x. 4. Solve 1 cos 2 sin 0x x− − = for 0 2x π≤ ≤ . 5. Solve using a calculator: sin 3 cos 2 0x x− = , 2 4xπ π− ≤ ≤

Homework: p.802 – (15-39)/3, 47-55 all