Download Reciprocal Trigonometric Functions

CC-31 Reciprocal
Trigonometric Functions
Common Core State Standards
MACC.912.F-IF.3.7e Graph . . . trigonometric functions,
showing period, midline, and amplitude.
MP 1, MP 2, MP 3, MP 4, MP 5
Objectives To evaluate reciprocal trigonometric functions
To graph reciprocal trigonometric functions
This asks only
for the length of
the extension, not
the length of the
extension ladder.
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.
5 ft
20 ft
MATHEMATICAL
PRACTICES
Lesson
Vocabulary
•cosecant
•secant
•cotangent
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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.
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.
1
1
sec u = cos
csc u = sin u
u
(cot u = 0 at odd multiples of p2 , where tan u is undefined.)
1
cot u = tan
u
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 =
1
y , sec u
=
1
x , cot u
=
x
y.
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Reciprocal Trigonometric Functions
CC-31 Reciprocal Trigonometric Functions
y P (x, y)
u
O 1
x
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You can use what you know about the unit circle to find exact values for reciprocal
trigonometric functions.
Problem 1 Finding Values Geometrically
(
)
( )
P
What are the exact values of cot − 5P
6 and csc 6 ? Do not use a calculator.
Find the point where the
unit circle intersects the
terminal side of the angle
- 5p
6 radians.
( 5p6 )
cot -
y
x
(– √23 , – 21(
Find the exact value of
cot - 5p
6 .
( )
(
cot −
–5p
6
5p
x
=y
6
)
23
− 2
=
(
cot −
Find the point where the
unit circle intersects the
terminal side of the angle
p
6 radians.
Find the exact value of
csc p6 .
( )
csc
− 21
5p
= 13
6
)
( p6 )
y
(√32 , 12 )
x
csc
π
6
( p6 ) = y1
=
csc
= 23
1
1
2
=2
( p6 ) = 2
Got It? 1. What is the exact value of each expression? Do not use a calculator.
p
a. csc 3
( )
5p
b. cot - 4
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.
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y
n
x
3
5
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Use the reciprocal relationships to evaluate secant, cosecant, or cotangent on a
calculator, since most calculators do not have these functions as menu options.
Problem 2 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
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.
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B cot 10
1
1
sec 2 = cos 2
1/cos(2)
cot 10 = tan 10
1/tan(10)
–2.402997962
sec 2 ≈ -2.403
1.542351045
cot 10 ≈ 1.542
C csc 35°
D cot P
csc 35° =
1
sin 35°
cot p = tan1 p
To evaluate an angle in degrees
in radian mode, use the degree
symbol from the ANGLE menu.
1/sin(35˚)
ERR:DIVIDE BY 0
1:Quit
2:Goto
1.743446796
Evaluating cot p results in an
error message, because tan p
is equal to zero.
csc 35° ≈ 1.743
Got It? 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°
3p
d. sec 2
e. Reasoning How can you find the cotangent of an angle without using the tangent
key on your calculator?
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Reciprocal Trigonometric Functions
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The graphs of reciprocal trigonometric functions have asymptotes where the
functions are undefined.
Problem 3 Sketching a Graph
For what values is
csc x undefined?
Wherever sin x = 0, its
reciprocal is undefined.
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.
x
0
sin x 0 0.5
csc x — 2
Step 2
p
3
p
2
2p
3
0.9
1.2
1
1
0.9 0.5 0 0.5 0.9 1 0.9 0.5 0
1.2 2 — 2 1.2 1 1.2 2 —
p
6
5p
6
p
7p
6
4p
3
3p
2
5p
3
11p
2p
6
Plot the points and sketch the graphs.
2
y csc x
1
O
1
y = csc x will have a vertical
asymptote whenever its
denominator (sin x) is 0.
y
y sin x p
x
2
Got It? 3. What are the graphs of y = tan x and y = cot x in the interval from 0 to 2p?
You can use a graphing calculator to graph trigonometric functions quickly.
Problem 4 Using Technology to Graph a Reciprocal Function
How can you find the
value?
Use the table feature of
your calculator.
Graph y = sec x. What is the value of sec 20°?
Step 1
Step 2
Use degree mode.
1
Graph y = cos
x.
Xmin –360
Xmax 360
Xscl 30
Ymin –5
Ymax 5
Yscl 1
Use the TABLE feature.
sec 20° ≈ 1.0642
X
20
21
22
23
24
25
26
X = 20
Y1
1.0642
1.0711
1.0785
1.0864
1.0946
1.1034
1.1126
Got It? 4. What is the value of csc 45°? Use the graph of the reciprocal
trigonometric function.
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You can use a reciprocal trigonometric function to solve a real-world problem.
Problem 5 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 d
of 40° and 70°?
U
601 ft
Set your calculator to degree mode.
Enter the function and construct
a table that gives values of d for
various angles of u.
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.
Plot1
Plot2
\Y1 = 601/cos(X)
\Y2 =
\Y3 =
\Y4 =
\Y5 =
\Y6 =
\Y7 =
Plot3
TABLE SETUP
TblStart = 20
Tbl = 10
Indpnt: Auto Ask
Depend: Auto Ask
X
20
30
40
50
60
70
80
Y1
639.57
693.98
784.55
934.99
1202
1757.2
3461
X = 20
From the table, the objects are about 785 feet away and 1757 feet away, respectively.
Got It? 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°?
Lesson Check
Do you know HOW?
Do you UNDERSTAND?
Find each value without using a calculator.
p
1. csc 2
p
2. sec 1- 6 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.
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MATHEMATICAL
PRACTICES
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?
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Practice and Problem-Solving Exercises
A
Practice
MATHEMATICAL
PRACTICES
See Problem 1.
Find each value without using a calculator. If the expression is undefined,
write undefined.
9. sec ( -p)
( 3p )
13. cot - 2
( p)
( )
5p
p
11. cot - 3
3p
15. sec - 4
10. csc 4
7p
14. csc 6
12. sec 2
16. cot ( -p)
See Problem 2.
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°
21. cot ( -32°)
22. sec 195°
23. csc 0
( 3p )
20. sec - 2
24. cot ( -0.6)
See Problem 3.
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°?
B
Apply
See Problem 4.
See Problem 5.
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 = sin1 x
1
41. y = cos
x
a.
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b.
42. y = - sin1 x
c.
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Graph each function in the interval from 0 to 2P.
p
1
u
45. y = -sec pu
43. y = csc u - 2
44. y = sec 4 u
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?
46. y = cot 3
48. Reasoning Use the relationship csc x = sin1 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
y
is 35 ft from the building. The function y = 35 sec u
models the length y in feet that the fire ladder must be
35 ft
to reach the building.
8 ft
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.)
θ
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
57. f (x) = 3 csc (x + 2) - 1
55. y = sec 2 1 u + 2 2
58. y = cot 2(x + p) + 3
p
56. y = -2 sec (x - 4)
59. g (x) = 2 sec 1 3 1 x - 6 22 - 2
p
Graph y = -cos x and y = -sec x on the same axes.
State the domain, range, and period of each function.
For which values of x does -cos x = -sec x? Justify your answer.
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.
60. a.
b.
c.
d.
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61. a. Reasoning Which expression gives the correct value of csc 60°?
I. sin ((60-1)°)
C
Challenge
II. (sin 60°)-1
III. (cos 60°)-1
1 °
b. Which expression in part (a) represents sin 1 60
2?
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.
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