algebra 2 honors: trigonometry€¦ · algebra 2 honors: trigonometry semester 2, unit 6: activity...

Algebra 2 Honors: Trigonometry Semester 2, Unit 6: Activity 31

Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 6

Vocabulary:

Arc length

Unit circle

Radian

Standard position

Initial side

Terminal side

Coterminal angles

Reference angle

Trigonometric function

Periodic function

Period

Amplitude

Midline

Phase shift

Unit Overview

In this unit, students build on their knowledge of trigonometry from geometry and extend it to radian measure and the unit circle. Students will apply trigonometric functions to understanding real-world periodic phenomena.

Student Focus

Main Ideas for success in lessons 31-1 & 31-2

Introduce students to radian measurement

Use a real-world problem to develop understanding of radian measure and how it differs from degree measure

Example:

Lesson 31-1: Vocabulary:

The arc length is the length of a portion of the circumference of a circle. The arc length is determined by the radius of the circle and by the angle measure that defines the corresponding arc, or portion, of the circumference.

When you find the arc length generated by a radius on a circle with radius 1, it is called a unit circle. On a unit circle, the constant of proportionality is the measure of

the angle of rotation written in radians, which equals the length of the corresponding arc on the unit circle.

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Example A: A toy train travels 30° around a circular track with a radius of 8 feet. What is the constant of proportionality that can be used to find the distance along the track that the train travels?

a)

b)

c)

d)

Convert radians to degrees multiply radians by the ratio

Convert degrees to radians multiply degree by the ratio

Lesson 31-2: Example A: A Ferris wheel makes one complete rotation every 4 minutes. How far, to the nearest tenth, will a rider who is seated 40 feet from the center travel in 10 minutes?

a) 15.7 feet b) 200 feet c) 251.3 feet

d) 628.3 feet

Example B: What is 150° in radians?

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Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 6

Vocabulary:

Arc length

Unit circle

Radian

Standard position

Initial side

Terminal side

Coterminal angles

Reference angle


Periodic function

Period

Amplitude

Midline

Phase shift

Unit Overview


Student Focus

Main Ideas for success in lessons 32-1 & 32-2

Calculate trigonometric ratios for acute angles using the ratios of the sides of a right triangle

Use reference angles and the unit circle to find trigonometric ratios of any angle

Example:

Lesson 32-1: Vocabulary: An angle is in standard position when the vertex is placed at the origin and the initial side is on the positive x-axis. The other ray that forms the angle is the terminal side.

Draw an angle in standard position with a measure of 120°.

Since 120° is 30° more than 90°, the terminal side is 30° counterclockwise from the positive y-axis.

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Draw an angle in standard position with a measure of -200°.

Since -200° is negative, the terminal side of 200° clockwise from the positive x-axis.

Draw and angle in standard position with a measure of

radians.

Since

is greater than radians, the terminal side makes one full rotation, plus an

additional

radians.

Find one positive and one negative angle that are coterminal with each given angle.

If θ is an angle in standard position, its reference angle α is the acute angle formed by the terminal side of θ and the x-axis. The graphs show the reference angle α for four different angles that have their terminal sides in different quadrants.

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The relationship between θ and α is shown for each quadrant when 0° < θ < 360° or 0 < θ < 2π.

Quadrant I Quadrant II Quadrant III Quadrant IV

QUESTION: Find the reference angle for . ANSWER: The terminal side of θ lies in Quadrant III, therefore so .

QUESTION: Find the reference angle for

.

When an angle is not between 0 and 360° ( ), find a coterminal angle that is within that range. Then use the coterminal angle to find the reference angle.

ANSWER: The terminal side of θ lies in Quadrant II, therefore

so

.

QUESTION: Find the reference angle for . Since 435° is greater than 360°, subtract . Now determine the reference angle for 75°. ANSWER: Since 75° is in Quadrant I, the reference angle is 75°.

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QUESTION: Find the reference angle for

radians.

Since

is greater than , subtract

ANSWER: The terminal side of this angle is in Quadrant III so

,

so

.

QUESTION: Find the sine and cosine of 90°

QUESTION: Find the sine and cosine of 180°.

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Lesson 32-2:

QUESTION: What are the sine and cosine of θ when ?

The sine and cosine are the lengths of the legs of a triangle.

If θ is not in the first quadrant, use a reference angle. ANSWER:

length of shorter leg

length of longer leg

QUESTION: What are sinθ and cosθ when

radians?

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Unit Circle:

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Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 6

Vocabulary:

Arc length

Unit circle

Radian

Standard position

Initial side

Terminal side

Coterminal angles

Reference angle


Periodic function

Period

Amplitude

Midline

Phase shift

Unit Overview


Student Focus

Main Ideas for success in lessons 33-1 & 33-2:

Use Pythagorean Theorem to prove the Pythagorean Identity,

Use the Pythagorean Identity to find given the value of

one of those functions and the quadrant of θ.

Combine the Pythagorean Identity with the reciprocal identities to prove related Pythagorean identities.

Example:

Lesson 33-1:

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EXAMPLE:

Lesson 33-2:

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Algebra 2 Honors: Trigonometry Semester 2, Unit 6: Activities 34 & 35

Resources:

SpringBoard-

Algebra 2

Online

Resources:

Algebra 2

Springboard Text

Unit 6

Vocabulary:

Arc length

Unit circle

Radian

Standard position

Initial side

Terminal side

Coterminal angles

Reference angle


Periodic function

Period

Amplitude

Midline

Phase shift

Unit Overview In this unit, students build on their knowledge of trigonometry from geometry and extend it to radian measure and the unit circle. Students will apply trigonometric functions to understanding real-world periodic phenomena.

Student Focus

Main Ideas for success in lessons 34-1, 34-2, 34-3, 34-4, & 34-5:

Analyze, graph, and write equations for parent trigonometric functions and their transformations, including phase shifts

Identify period, midline, amplitude, and asymptotes

Main Idea for success in lesson 35-1:

Using knowledge of trigonometric functions and their graphs, model real-world periodic phenomena using functions of the form or

Example:

Lesson 34-1:

Example 1:

Period: 3

Amplitude: 4

Midline: y = 2

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Lesson 34-2: Graph of Sine Function

Transformations of Sine Function

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Lesson 34-3:

Graph of Cosine Function

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Transformations of Cosine Functions

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Lesson 34-4:

Graph of Tangent Function

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Transformations of Tangent Function

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Lesson 34-5:

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Summary of Transformations

y = a sin b(x h) + k y = a cos b(x h) + k y = a tan b(x h) + k

a The coefficient changes the amplitude of the sine and cosine functions.

When a > 1, the amplitude increases and the graph is stretched vertically.

When 0 < a < 1, the amplitude decreases and the graph is compressed vertically.

When a < 0, the graph is reflected across the x-axis.

b The coefficient changes the period.

When b > 1, the period decreases and the graph is compressed horizontally.

When 0 < b < 1, the period increases and the graph is stretched horizontally.

The period of sin bx and cos bx is 2πb.

The period of tan bx is πb.

h The constant shifts the graph horizontally.

When h > 0, the graph shifts to the right.

When h < 0, the graph shifts to the left.

k The constant shifts the graph vertically.

When k > 0, the graph shifts up.

When k < 0, the graph shifts down.

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Lesson 35-1:

Example 1: Hector’s skateboard wheels have a diameter of 80 millimeters. When he starts riding, a small chip in one wheel is visible at the top of the wheel. As he rides, the wheels make 6 revolutions per second. What is the function that gives the height h in millimeters of the chip in the wheel as a function of time t in seconds?

Example 2:

The tide at Lookout Point is modeled by

, giving

height h in feet as a function of time t in hours since low tide.

Which describes the heights of low and high tide and the time in between them?

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Name class date

1© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 6 Practice

LeSSon 31-1 1. Going around a circle you travel the length of an

arc formed by a 45° angle. If the distance you travel is 20 m, what is the radius of the circle?

2. Make sense of problems. Two people are sitting in different locations on a merry-go-round. The first person sits 4 m from the center of the ride and the second person sits 6 m from the center of the ride. How much farther has the second person traveled than the first person when they complete a turn?

3. Write the constant of proportionality for each of the following angles in a unit circle. Express your answers in terms of p.

a. 3°

b. 18°

c. 54°

d. 99°

4. Which of the following constants of proportionality in a unit circle is paired with its corresponding angle?

A. 119p , 20° B. 4

8p , 90°

C. 25p , 24° D. 14

9p , 280°

5. Reason quantitatively. A passenger sits 20 m from the center of a Ferris wheel and travels a quarter of a turn. What is an accurate estimate of the distance traveled in meters?

LeSSon 31-2 6. Convert the following angle measures from radians

to degrees.

a. 27p

b. 38p

c. 3.14159

d. 5

7. Make sense of problems. The measure of linear velocity is meters per second and the measure of angular velocity is radians per second. Angular velocity describes how fast an object in circular motion travels in terms of the angle measure of the arc (in radians) that the object travels in a second. If you are on a merry-go-round sitting 4 m from the center and you travel

9p

radians per second, what is your linear velocity? In other words, what is the length of the arc that you travel every second?

8. Model with mathematics. Come up with a general formula to convert angular velocity v to linear velocity v in terms of the radius r.

Algebra 2 Unit 6 Practice

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2

Name class date

© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 6 Practice

9. It takes a Ferris wheel 12 minutes to complete a turn. How many radians does it turn every minute?

A. 0.48 B. 0.52

C. 27p

D. 95p

10. A Frisbee rotates 40 times every two seconds. How many radians does it rotate every second?

A. 127p B. 20p

C. 40p D. 10p

LeSSon 32-1 11. Which of the following angle pairs are coterminal

angles?

A. p p

254

,54

B. 54

,4

p p2

C. 94

,34

p p2 D.

94

,74

p p2

12. Which of the following angle pairs are coterminal with 33°?

A. 213°, 22147° B. 7130

,4930

p p2

C. 393°, 2327° D. p p

27160

, 4960

13. Reason quantitatively. Are angles measuring

88° and 6845

p2 radians coterminal? Explain your

answer.

14. Make sense of problems. Imagine looking at a Ferris wheel from the side so that we can divide it into four quadrants. A passenger starts at the very bottom of the Ferris wheel, and the Ferris wheel makes a complete turn counterclockwise every 10 minutes. What quadrant will the passenger be in at 14 minutes? Explain.

15. What are the reference angles for the following angles?

a. 78°

b. 268°

c. 38p

LeSSon 32-2 16. Find the sine of each of the following angles.

a. 585°

b. 750°

c. 240°

17. Find the cosine of each of the following angles.

a. 585°

b. 750°

c. 240°

18. What is the tangent of 480°?

A. 1 B. 0

C. 3 D. 2 3

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19. Reason abstractly. The sum of the internal angles of a triangle is 180°. If a right triangle has one angle that is 45°, what can be deduced about the length of the two shorter sides of the triangle?

20. express regularity in repeated reasoning. Complete the following table.

0° 30° 45° 60° 90°

sine 02

12 2 2 2

cosine 42

32 2 2 2

tangent

LeSSon 33-1 21. Given that 0 , u ,

2,p find the value of cos u for

each of the given values of sin u.

a. sin u 5 58

b. sin u 5 14

c. sin u 5 37

22. Given that 0 , u , 2,p find the value of sin u for

each of the given values of cos u.

a. cos u 5 23

b. cos u 5 17

c. cos u 5 47

23. Given that sin u 5 38

, and 0 , u , 2,p what are

the values of cos u and tan u?

A. cos u 5 558

and tan u 5 3 55

55

B. cos u 5 2558

and tan u 5 23 55

55

C. cos u 5 2 58

and tan u 5 255 3

3

D. cos u 5 22 5

8 and tan u 5

55 33

24. Reason abstractly. Why can the value of the tangent function exceed 1 when the values of the sine and cosine functions cannot?

25. Make sense of problems. A right triangle has a hypotenuse of length 8 m. If one of the angles of the triangle is 30°, what are the lengths of the other two sides of the triangle?

LeSSon 33-2 26. Simplify the following expression.

csc u sin u 1 cot2 u

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27. Simplify the following expression. sin2 u 2 tan u cot u

28. Given that sin u 5 23

, 0 , u , 2

,p what are the

values of cos u and sec u?

A. cos u 5 5

3 and sec u 5

35

B. cos u 5 3 5

5 and sec u 5

53

C. cos u 5 3 5

52 and sec u 5

53

2

D. cos u 5 5

32 and sec u 5

3 55

2

29. Reason abstractly. The cosecant, secant, and tangent functions are the reciprocals of the sine, cosine, and tangent functions respectively. What is always the product of the multiplication of each of these functions by its reciprocal?

30. Make sense of problems. Does sin2 x 2 (cos x) (sec x) 5 sin2 x 2 1? Explain your answer.

LeSSon 34-1 31. Is this graph a periodic function? If so, find the

amplitude. The maximum value of the function is 7 and the minimum value is 27.

x

y

25210

210

25

5

10

5 10

32. Is this graph a periodic function? If so, find the amplitude. The maximum value of the function is 39 and the minimum value is 25.

x

y

225250

250

225

25

50

25 50

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33. Reason qualitatively. Is the function represented by the graph below a periodic function? Why or why not?

x

y

25210

210

25

5

10

5 10

34. Attend to precision. Estimate the period and the amplitude for the function graphed below.

x

y

25210

210

25

5

10

5 10

35. What is the maximum value of the function f(x) 5 4 cos (x)?

A. ` B. 0

C. 4 D. 1

LeSSon 34-2 36. Use appropriate tools strategically. Complete

the table below and use it to construct a graph of the function y 5 3 sin x.

x sin x y 5 3 sin x

0

3p

2p

23p

p

43p

32p

53p

2p

37. What is the amplitude, midline and period of the

function y 5 38

sin x?

38. What is the sine function that has an amplitude of56

, a midline of y 5 0, and a period of 4p

?

39. Which of the following sine functions describes a graph that has a period of 8p, an amplitude of 2, and a midline of y 5 1?

A. y 5 2 sin (4x) 1 1

B. y 5 2 sin (2x) 1 1

C. y 5 2 sin x4

1 1

D. y 5 2 sin x2

1 1

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40. Attend to precision. Find the value of y when

x 5 3p

in the function y 5 4 sin2 x2

1 3.

LeSSon 34-3 41. Attend to precision. Find the value of y when

x 5 4p in the function y 5 4 cos2 x

16

1 2.

42. Write the equation for the cosine function that has

an amplitude of 23

, a period of 2p, and a midline of

y 5 0.

43. What are the amplitude, midline, and period of the

function y 5 15

cos 15

x?

44. Which are the maximum and minimum values for

the function f (x) 5 23

cos (x) 1 1?

A. maximum 5 23

, minimum 5 223

B. maximum 5 53

, minimum 5 13

C. maximum 5 53

, minimum 5 213

D. maximum 5 23

, minimum 5 213

45. Use appropriate tools strategically. Complete the table below and use it to construct a graph of

the function y 5 23

cos x.

x cos x y 5 2

3 cos x

0

3p

2p

23p

p

43p

32p

53p

2p

LeSSon 34-4 46. What is the range of the function

y 5 3 tan (2x) 1 1?

47. Reason qualitatively. Given that tan x 5 sincos

xx

and that the domains for both sin x and cos x are

all real numbers, why are there gaps in the graph for y 5 tan x?


x 5 4p in the function y 5 5 tan2 x

16

1 2.

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49. Name the period, zeros, and asymptotes of the

function y 5 45

tan 37

x.

50. Which of the following is not a zero for the

function y 5 tan 23

x?

A. x 5 23p

B. x 5 32p

C. x 5 32p

2 D. x 5 0

LeSSon 34-5 51. List the amplitude and period and describe the

horizontal and vertical shifts relative to the parent

function of y 5 3 sin 4

x

6p

1 2 1.

52. List the amplitude and period and describe the horizontal and vertical shifts relative to the parent

function of y 5 35

cos 2

x 2

3p

2 1 4.

53. Which are the features of the graph of

y 5 52

tan 3

x 3

2p

2 1 4?

A. amplitude: 52

, period: 3

,p horizontal shift:

32p right, vertical shift: 4 up

B. period: 3


32p

right,

vertical shift: 4 up

C. amplitude: 52

, period: 3


32p

left, vertical shift: 4 up

D. period: 3


32p

right,

vertical shift: 4 down

54. Reason abstractly. The function y 5 cot x is the reciprocal of the function y 5 tan x. Describe how the graph of y 5 cot x compares to the graph of y 5 tan x.


x 5 4p in the function y 5 4 cos2 x4

1 sin2 x

1 2 tan2 x2

.

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LeSSon 35-1 56. Write a trigonometric function that describes the

height as a function of time of a car on a Ferris wheel that makes a complete rotation every 2 minutes. The radius of the Ferris wheel is 25 m and at its highest point the car is 60 m high.

57. Attend to precision. What is the height of the car in Item 56 after 7 minutes and 12 seconds?

58. Make sense of problems. What is the height of the car in Item 56 at its lowest point?

59. A car engine is running at 6000 revolutions (turns) per minute. Which of the following could describe the position of a point on a gear attached directly to the engine as a function of time t in seconds?

A. h(t) 5 0.3 sin

t

6000p

1 1

B. h(t) 5 0.3 sin (200pt)

C. h(t) 5 0.3 sin

t

606000

p 1 1

D. h(t) 5 0.3 sin

t

6000p

60. The distance of an object from the ground in meters can be expressed as a function of time in seconds

by the function h(t) 5 7 cos

t

27p

1 400. What is

the maximum height of this object, and when does the object reach that height for the first time?

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algebra 2 honors: trigonometry€¦ · algebra 2 honors: trigonometry semester 2, unit 6: activity...

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