Common Core State Standards CC-31 P Reciprocal Then sketch ...atmshs.enschool.org/ourpages/auto/2015/10/18... · C Challenge HSM15_A2Hon_SE_CC_30_TrKit.indd 150 8/5/13 7:19 PM Lesson

150 Chapter 13 Periodic Functions and Trigonometry

Use a graphing calculator to graph each function in the interval from 0 to 2P. Then sketch each graph.

49. y = sin x + x 50. y = sin x + 2x

51. y = cos x - 2x 52. y = cos x + x

53. y = sin (x + cos x) 54. y = sin (x + 2 cos x)

ChallengeC

HSM15_A2Hon_SE_CC_30_TrKit.indd 150 8/5/13 7:19 PM

Lesson 13-8 Reciprocal Trigonometric Functions 151

Reciprocal Trigonometric FunctionsObjectives To evaluate reciprocal trigonometric functions To graph reciprocal trigonometric functions

To solve an equation ax = b, you multiply each side by the reciprocal of a. If a is a trigonometric expression, you need to use its reciprocal.

Essential Understanding Cosine, sine, and tangent have reciprocals. Cosine and secant are reciprocals, as are sine and cosecant. Tangent and cotangent are also reciprocals.

Lesson Vocabulary

•cosecant•secant•cotangent

LessonVocabulary

Key Concept Cosecant, Secant, and Cotangent Functions

The cosecant (csc), secant (sec), and cotangent (cot) functions are defined using reciprocals. Their domains do not include the real numbers u that make the denominator zero.

csc u = 1sin u sec u = 1

cos u cot u = 1tan u

(cot u = 0 at odd multiples of p2 , where tan u is undefined.)

You want the extension ladder to reach the window sill so you can wash the top window. What expression gives the length by which you should extend the ladder while keeping the base in place? Explain.This asks only

for the length of the extension, not the length of the extension ladder.

You can use the unit circle to evaluate the reciprocal trigonometric functions directly. Suppose the terminal side of an angle u in standard position intersects the unit circle at the point (x, y).

Then csc u = 1y , sec u = 1

x , cot u = xy .

x1

y

u

O

P (x, y)

CC-31

MATHEMATICAL PRACTICES

20 ft

5 ft

70�

MACC.912.F-IF.3.7e Graph . . . trigonometric functions, showing period, midline, and amplitude.

MP 1, MP 2, MP 3, MP 4, MP 5

Common Core State Standards

HSM15_A2Hon_SE_CC_31_TrKit.indd 151 8/5/13 7:20 PMCC-31 Reciprocal Trigonometric Functions 151

Problem 1

Got It?


You can use what you know about the unit circle to find exact values for reciprocal trigonometric functions.

Finding Values Geometrically

What are the exact values of cot (−5P6 ) and csc (P6 )? Do not use a calculator.

cot (-5p6 )

y

x

√32

12( (, –5p

6– –

cot (−5p6 ) = x

y

=− 23

2− 1

2= 23

cot (−5p6 ) = 13

csc (p6 )

csc (p6 ) = 1y

=112= 2

csc (p6 ) = 2

x

y12( )

π6

√32

,

Find the exact value of cot (- 5p6 ).

Find the point where the unit circle intersects the terminal side of the angle p6 radians.

Find the exact value of csc ( p6 ).

Find the point where the unit circle intersects the terminal side of the angle - 5p6 radians.

1. What is the exact value of each expression? Do not use a calculator.

a. csc p3 b. cot (-5p4 ) c. sec 3p

d. Reasoning Use the unit circle at the right to find cot n, csc n, and sec n. Explain how you found your answers.

xn

y

35


Got It?

Problem 2


Use the reciprocal relationships to evaluate secant, cosecant, or cotangent on a calculator, since most calculators do not have these functions as menu options.

Finding Values with a Calculator

What is the decimal value of each expression? Use the radian mode on your calculator. Round to the nearest thousandth.

A sec 2 B cot 10

sec 2 = 1cos 2 cot 10 = 1

tan 10

sec 2 ≈ -2.403 cot 10 ≈ 1.542

C csc 35° D cot P

csc 35° = 1sin 35° cot p = 1

tan p

To evaluate an angle in degrees in radian mode, use the degree symbol from the ANGLE menu.

csc 35° ≈ 1.743

2. What is the decimal value of each expression? Use the radian mode on your calculator. Round your answers to the nearest thousandth.

a. cot 13 b. csc 6.5 c. sec 15° d. sec 3p2 e. Reasoning How can you find the cotangent of an angle without using the tangent

key on your calculator?

1/cos(2) –2.402997962

1/tan(10) 1.542351045

1/sin(35˚) 1.743446796

ERR:DIVIDE BY 01:Quit2:Goto

Evaluating cot p results in an error message, because tan p is equal to zero.

Can you use the sin−1, cos−1, and tan−1 keys on the calculator for the reciprocal functions? No; those keys are inverse functions, not reciprocal functions.

HSM15_A2Hon_SE_CC_31_TrKit.indd 153 8/5/13 7:20 PM152 Common Core

Problem 1

Got It?


You can use what you know about the unit circle to find exact values for reciprocal trigonometric functions.

Finding Values Geometrically

What are the exact values of cot (−5P6 ) and csc (P6 )? Do not use a calculator.

cot (-5p6 )

y

x

√32

12( (, –5p

6– –

cot (−5p6 ) = x

y

=− 23

2− 1

2= 23

cot (−5p6 ) = 13

csc (p6 )

csc (p6 ) = 1y

=112= 2

csc (p6 ) = 2

x

y12( )

π6

√32

,

Find the exact value of cot (- 5p6 ).

Find the point where the unit circle intersects the terminal side of the angle p6 radians.

Find the exact value of csc ( p6 ).

Find the point where the unit circle intersects the terminal side of the angle - 5p6 radians.

1. What is the exact value of each expression? Do not use a calculator.

a. csc p3 b. cot (-5p4 ) c. sec 3p

d. Reasoning Use the unit circle at the right to find cot n, csc n, and sec n. Explain how you found your answers.

xn

y

35


Got It?

Problem 2


Use the reciprocal relationships to evaluate secant, cosecant, or cotangent on a calculator, since most calculators do not have these functions as menu options.

Finding Values with a Calculator

What is the decimal value of each expression? Use the radian mode on your calculator. Round to the nearest thousandth.

A sec 2 B cot 10

sec 2 = 1cos 2 cot 10 = 1

tan 10

sec 2 ≈ -2.403 cot 10 ≈ 1.542

C csc 35° D cot P

csc 35° = 1sin 35° cot p = 1

tan p

To evaluate an angle in degrees in radian mode, use the degree symbol from the ANGLE menu.

csc 35° ≈ 1.743

2. What is the decimal value of each expression? Use the radian mode on your calculator. Round your answers to the nearest thousandth.

a. cot 13 b. csc 6.5 c. sec 15° d. sec 3p2 e. Reasoning How can you find the cotangent of an angle without using the tangent

key on your calculator?

1/cos(2) –2.402997962

1/tan(10) 1.542351045

1/sin(35˚) 1.743446796

ERR:DIVIDE BY 01:Quit2:Goto

Evaluating cot p results in an error message, because tan p is equal to zero.

Can you use the sin−1, cos−1, and tan−1 keys on the calculator for the reciprocal functions? No; those keys are inverse functions, not reciprocal functions.


Problem 4

Got It?

Got It?

Problem 3


You can use a graphing calculator to graph trigonometric functions quickly.

Using Technology to Graph a Reciprocal Function

Graph y = sec x. What is the value of sec 20°?

Step 1 Use degree mode. Step 2 Use the TABLE feature. Graph y = 1

cos x. sec 20° ≈ 1.0642

4. What is the value of csc 45°? Use the graph of the reciprocal trigonometric function.

Xmin � –360Xmax � 360Xscl � 30Ymin � –5Ymax � 5Yscl � 1

X=20

X Y120212223242526

1.06421.07111.07851.08641.09461.10341.1126

How can you find the value?Use the table feature of your calculator.

The graphs of reciprocal trigonometric functions have asymptotes where the functions are undefined.

Sketching a Graph

What are the graphs of y = sin x and y = csc x in the interval from 0 to 2P?

Step 1 Make a table of values.

Step 2 Plot the points and sketch the graphs. y = csc x will have a vertical

asymptote whenever its denominator (sin x) is 0.

3. What are the graphs of y = tan x and y = cot x in the interval from 0 to 2p?

x

sin x

csc x

0 p 2pp6

p3

p2

2p3

5p6

7p6

4p3

3p2

5p3

11p6

0 0.5 0.9 1 0.9 0.5 0 �0.5 �0.9 �1 �0.9 �0.5 0— 2 1.2 1 1.2 2 — �2 �1.2 �1 �1.2 �2 —

�2

�1

1

2

x

y

O p

y � csc x

y � sin x

For what values is csc x undefined?Wherever sin x = 0, its reciprocal is undefined.


601 ftd

601 ftd

UU

Problem 5

Got It?


Lesson CheckDo you know HOW?Find each value without using a calculator.

1. csc p2 2. sec 1-p6 2

Use a calculator to find each value. Round your answers to the nearest thousandth.

3. csc 1.5 4. sec 42°

5. An extension ladder leans against a building forming a 50° angle with the ground. Use the function y = 21 csc x + 2 to find y, the length of the ladder. Round to the nearest tenth of a foot.

Do you UNDERSTAND? 6. Reasoning Explain why the graph of y = 5 sec u has

no zeros.

7. Error Analysis On a quiz, a student wrote sec 20° + 1 = 0.5155. The teacher marked it wrong. What error did the student make?

8. Compare and Contrast How are the graphs of y = sec x and y = csc x alike? How are they different? Could the graph of y = csc x be a transformation of the graph of y = sec x?

You can use a reciprocal trigonometric function to solve a real-world problem.

Using Reciprocal Functions to Solve a Problem

A restaurant is near the top of a tower. A diner looks down at an object along a line of sight that makes an angle of U with the tower. The distance in feet of an object from the observer is modeled by the function d = 601 sec U. How far away are objects sighted at angles of 40° and 70°?

Set your calculator to degree mode. Enter the function and construct a table that gives values of d for various angles of u.

From the table, the objects are about 785 feet away and 1757 feet away, respectively.

5. The 601 in the function for Problem 5 is the diner’s height above the ground in feet. If the diner is 553 feet above the ground, how far away are objects sighted at angles of 50° and 80°?

Plot1 Plot2 Plot3\Y1\Y2\Y3\Y4\Y5\Y6\Y7

= 601/cos(X)= = = = = =

TABLE SETUP TblStart = 20 Tbl = 10Indpnt: Auto Ask Depend: Auto Ask

639.57693.98784.55934.9912021757.23461

X Y120304050607080

X=20

How can you check that your answers are correct?Multiply the answers by cos u. If the answers are correct, then the product is 601.



Problem 4

Got It?

Got It?

Problem 3


You can use a graphing calculator to graph trigonometric functions quickly.

Using Technology to Graph a Reciprocal Function

Graph y = sec x. What is the value of sec 20°?

Step 1 Use degree mode. Step 2 Use the TABLE feature. Graph y = 1

cos x. sec 20° ≈ 1.0642

4. What is the value of csc 45°? Use the graph of the reciprocal trigonometric function.

Xmin � –360Xmax � 360Xscl � 30Ymin � –5Ymax � 5Yscl � 1

X=20

X Y120212223242526

1.06421.07111.07851.08641.09461.10341.1126

How can you find the value?Use the table feature of your calculator.

The graphs of reciprocal trigonometric functions have asymptotes where the functions are undefined.

Sketching a Graph

What are the graphs of y = sin x and y = csc x in the interval from 0 to 2P?

Step 1 Make a table of values.

Step 2 Plot the points and sketch the graphs. y = csc x will have a vertical

asymptote whenever its denominator (sin x) is 0.

3. What are the graphs of y = tan x and y = cot x in the interval from 0 to 2p?

x

sin x

csc x

0 p 2pp6

p3

p2

2p3

5p6

7p6

4p3

3p2

5p3

11p6

0 0.5 0.9 1 0.9 0.5 0 �0.5 �0.9 �1 �0.9 �0.5 0— 2 1.2 1 1.2 2 — �2 �1.2 �1 �1.2 �2 —

�2

�1

1

2

x

y

O p

y � csc x

y � sin x

For what values is csc x undefined?Wherever sin x = 0, its reciprocal is undefined.


601 ftd

601 ftd

UU

Problem 5

Got It?


Lesson CheckDo you know HOW?Find each value without using a calculator.

1. csc p2 2. sec 1-p6 2

Use a calculator to find each value. Round your answers to the nearest thousandth.

3. csc 1.5 4. sec 42°

5. An extension ladder leans against a building forming a 50° angle with the ground. Use the function y = 21 csc x + 2 to find y, the length of the ladder. Round to the nearest tenth of a foot.

Do you UNDERSTAND? 6. Reasoning Explain why the graph of y = 5 sec u has

no zeros.

7. Error Analysis On a quiz, a student wrote sec 20° + 1 = 0.5155. The teacher marked it wrong. What error did the student make?

8. Compare and Contrast How are the graphs of y = sec x and y = csc x alike? How are they different? Could the graph of y = csc x be a transformation of the graph of y = sec x?

You can use a reciprocal trigonometric function to solve a real-world problem.

Using Reciprocal Functions to Solve a Problem

A restaurant is near the top of a tower. A diner looks down at an object along a line of sight that makes an angle of U with the tower. The distance in feet of an object from the observer is modeled by the function d = 601 sec U. How far away are objects sighted at angles of 40° and 70°?

Set your calculator to degree mode. Enter the function and construct a table that gives values of d for various angles of u.

From the table, the objects are about 785 feet away and 1757 feet away, respectively.

5. The 601 in the function for Problem 5 is the diner’s height above the ground in feet. If the diner is 553 feet above the ground, how far away are objects sighted at angles of 50° and 80°?

Plot1 Plot2 Plot3\Y1\Y2\Y3\Y4\Y5\Y6\Y7

= 601/cos(X)= = = = = =

TABLE SETUP TblStart = 20 Tbl = 10Indpnt: Auto Ask Depend: Auto Ask

639.57693.98784.55934.9912021757.23461

X Y120304050607080

X=20

How can you check that your answers are correct?Multiply the answers by cos u. If the answers are correct, then the product is 601.




Practice and Problem-Solving Exercises

Find each value without using a calculator. If the expression is undefined, write undefined.

9. sec (-p) 10. csc 5p4 11. cot (-p3 ) 12. sec p2

13. cot (-3p2 ) 14. csc 7p6 15. sec (-3p

4 ) 16. cot (-p)

Graphing Calculator Use a calculator to find each value. Round your answers to the nearest thousandth.

17. sec 2.5 18. csc (-0.2) 19. cot 56° 20. sec (-3p2 )

21. cot (-32°) 22. sec 195° 23. csc 0 24. cot (-0.6)

Graph each function in the interval from 0 to 2P.

25. y = sec 2u 26. y = cot u 27. y = csc 2u - 1 28. y = csc 2u

Graphing Calculator Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places.

29. sec 30° 30. sec 80° 31. sec 110° 32. csc 30°

33. csc 70° 34. csc 130° 35. cot 30° 36. cot 60°

37. Distance A woman looks out a window of a building. She is 94 feet above the ground. Her line of sight makes an angle of u with the building. The distance in feet of an object from the woman is modeled by the function d = 94 sec u. How far away are objects sighted at angles of 25° and 55°?

38. Think About a Plan A communications tower has wires anchoring it to the ground. Each wire is attached to the tower at a height 20 ft above the ground. The length y of the wire is modeled with the function y = 20 csc u, where u is the measure of the angle formed by the wire and the ground. Find the length of wire needed to form an angle of 45°.

• Do you need to graph the function? • How can you rewrite the function so you can use a calculator?

39. Multiple Representations Write a cosecant model that has the same graph as y = sec u.

Match each function with its graph.

40. y = 1sin x 41. y = 1

cos x 42. y = - 1sin x

a. b. c.

PracticeA See Problem 1.

See Problem 2.

See Problem 3.

See Problem 4.

See Problem 5.

ApplyB





43. y = csc u - p2 44. y = sec 14 u 45. y = -sec pu 46. y = cot u3

47. a. What are the domain, range, and period of y = csc x? b. What is the relative minimum in the interval 0 … x … p? c. What is the relative maximum in the interval p … x … 2p?

48. Reasoning Use the relationship csc x = 1sin x to explain why each statement is true.

a. When the graph of y = sin x is above the x-axis, so is the graph of y = csc x. b. When the graph of y = sin x is near a y-value of -1, so is the graph of y = csc x.

Writing Explain why each expression is undefined.

49. csc 180° 50. sec 90° 51. cot 0°

52. Indirect Measurement The fire ladder forms an angle of measure u with the horizontal. The hinge of the ladder is 35 ft from the building. The function y = 35 sec u models the length y in feet that the fire ladder must be to reach the building.

a. Graph the function. b. In the photo, u = 13°. What is the ladder’s length? c. How far is the ladder extended when it forms an angle

of 30°? d. Suppose the ladder is extended to its full length of 80 ft. What

angle does it form with the horizontal? How far up a building can the ladder reach when fully extended? (Hint: Use the information in the photo.)

53. a. Graph y = tan x and y = cot x on the same axes. b. State the domain, range, and asymptotes of each function. c. Compare and Contrast Compare the two graphs. How are they alike? How are

they different? d. Geometry The graph of the tangent function is a reflection image of the

graph of the cotangent function. Name at least two reflection lines for such a transformation.

Graphing Calculator Graph each function in the interval from 0 to 2P. Describe any phase shift and vertical shift in the graph.

54. y = sec 2u + 3 55. y = sec 21u + p2 2 56. y = -2 sec (x - 4)

57. f (x) = 3 csc (x + 2) - 1 58. y = cot 2(x + p) + 3 59. g (x) = 2 sec 131x - p6 22 - 2

60. a. Graph y = -cos x and y = -sec x on the same axes. b. State the domain, range, and period of each function. c. For which values of x does -cos x = -sec x? Justify your answer. d. Compare and Contrast Compare the two graphs. How are they alike? How are

they different? e. Reasoning Is the value of -sec x positive when -cos x is positive and negative

when -cos x is negative? Justify your answer.

8 ft35 ft

θ

y



Practice and Problem-Solving Exercises

Find each value without using a calculator. If the expression is undefined, write undefined.

9. sec (-p) 10. csc 5p4 11. cot (-p3 ) 12. sec p2

13. cot (-3p2 ) 14. csc 7p6 15. sec (-3p

4 ) 16. cot (-p)

Graphing Calculator Use a calculator to find each value. Round your answers to the nearest thousandth.

17. sec 2.5 18. csc (-0.2) 19. cot 56° 20. sec (-3p2 )

21. cot (-32°) 22. sec 195° 23. csc 0 24. cot (-0.6)


25. y = sec 2u 26. y = cot u 27. y = csc 2u - 1 28. y = csc 2u

Graphing Calculator Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places.

29. sec 30° 30. sec 80° 31. sec 110° 32. csc 30°

33. csc 70° 34. csc 130° 35. cot 30° 36. cot 60°

37. Distance A woman looks out a window of a building. She is 94 feet above the ground. Her line of sight makes an angle of u with the building. The distance in feet of an object from the woman is modeled by the function d = 94 sec u. How far away are objects sighted at angles of 25° and 55°?

38. Think About a Plan A communications tower has wires anchoring it to the ground. Each wire is attached to the tower at a height 20 ft above the ground. The length y of the wire is modeled with the function y = 20 csc u, where u is the measure of the angle formed by the wire and the ground. Find the length of wire needed to form an angle of 45°.

• Do you need to graph the function? • How can you rewrite the function so you can use a calculator?

39. Multiple Representations Write a cosecant model that has the same graph as y = sec u.

Match each function with its graph.

40. y = 1sin x 41. y = 1

cos x 42. y = - 1sin x

a. b. c.

PracticeA See Problem 1.

See Problem 2.

See Problem 3.

See Problem 4.

See Problem 5.

ApplyB





43. y = csc u - p2 44. y = sec 14 u 45. y = -sec pu 46. y = cot u3

47. a. What are the domain, range, and period of y = csc x? b. What is the relative minimum in the interval 0 … x … p? c. What is the relative maximum in the interval p … x … 2p?

48. Reasoning Use the relationship csc x = 1sin x to explain why each statement is true.

a. When the graph of y = sin x is above the x-axis, so is the graph of y = csc x. b. When the graph of y = sin x is near a y-value of -1, so is the graph of y = csc x.

Writing Explain why each expression is undefined.

49. csc 180° 50. sec 90° 51. cot 0°

52. Indirect Measurement The fire ladder forms an angle of measure u with the horizontal. The hinge of the ladder is 35 ft from the building. The function y = 35 sec u models the length y in feet that the fire ladder must be to reach the building.

a. Graph the function. b. In the photo, u = 13°. What is the ladder’s length? c. How far is the ladder extended when it forms an angle

of 30°? d. Suppose the ladder is extended to its full length of 80 ft. What

angle does it form with the horizontal? How far up a building can the ladder reach when fully extended? (Hint: Use the information in the photo.)

53. a. Graph y = tan x and y = cot x on the same axes. b. State the domain, range, and asymptotes of each function. c. Compare and Contrast Compare the two graphs. How are they alike? How are

they different? d. Geometry The graph of the tangent function is a reflection image of the

graph of the cotangent function. Name at least two reflection lines for such a transformation.

Graphing Calculator Graph each function in the interval from 0 to 2P. Describe any phase shift and vertical shift in the graph.

54. y = sec 2u + 3 55. y = sec 21u + p2 2 56. y = -2 sec (x - 4)

57. f (x) = 3 csc (x + 2) - 1 58. y = cot 2(x + p) + 3 59. g (x) = 2 sec 131x - p6 22 - 2

60. a. Graph y = -cos x and y = -sec x on the same axes. b. State the domain, range, and period of each function. c. For which values of x does -cos x = -sec x? Justify your answer. d. Compare and Contrast Compare the two graphs. How are they alike? How are

they different? e. Reasoning Is the value of -sec x positive when -cos x is positive and negative

when -cos x is negative? Justify your answer.

8 ft35 ft

θ

y



61. a. Reasoning Which expression gives the correct value of csc 60°?

I. sin ((60-1)°) II. (sin 60°)-1 III. (cos 60°)-1

b. Which expression in part (a) represents sin 1 1602°?

62. Reasoning Each branch of y = sec x and y = csc x is a curve. Explain why these curves cannot be parabolas. (Hint: Do parabolas have asymptotes?)

63. Reasoning Consider the relationship between the graphs of y = cos x and y = cos 3x. Use the relationship to explain the distance between successive branches of the graphs of y = sec x and y = sec 3x.

64. a. Graph y = cot x, y = cot 2x, y = cot (-2x), and y = cot 12x on the same axes.

b. Make a Conjecture Describe how the graph of y = cot bx changes as the value of b changes.

ChallengeC


Documents

Common Core State Standards CC-31 P Reciprocal Then sketch ...atmshs.enschool.org/ourpages/auto/2015/10/18... · C Challenge HSM15_A2Hon_SE_CC_30_TrKit.indd 150 8/5/13 7:19 PM Lesson