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Lesson 4-2
Lesson
4-2
Vocabulary
Sines, Cosines, and
Tangents
unit circle
cosine, cos
sine, sin
circular function
The sine and cosine functions relate magnitudes of
rotations to coordinates of points on the unit circle.
BIG IDEA
trigonometric function
tangent, tan
The Sine, Cosine, and Tangent Functions
Mental Math
The unit circle is the circle with center at the
origin and radius 1, as shown at the right.
The following regular
polygons are inscribed in
a circle. Find the
measure of the angle
formed by two rays from
the center of the circle
that contain adjacent
vertices of the polygon.
Consider the rotation of magnitude θ
with center at the origin. We call this Rθ.
Regardless of the value of θ, the image of
(1, 0) under Rθ is on the unit circle. Call this
point P. We associate two numbers with each
value of θ. The cosine of θ (abbreviated
cos θ) is the x-coordinate of P; the sine of θ
(abbreviated sin θ) is the y-coordinate of P.
P = Rθ (1, 0) = (cos θ, sin θ)
y
P
θ
O
x
(1, 0)
a. triangle
b. square
c. pentagon d. octagon
Definition of cos θ and sin θ
For all real numbers θ, (cos θ, sin θ) is the image of the point
(1, 0) under a rotation of magnitude θ about the origin. That is,
(cos θ, sin θ) = Rθ (1, 0).
Because their definitions are based on a circle, the sine and cosine
functions are sometimes called circular functions of θ. They are also
called trigonometric functions, from the Greek word meaning “triangle
measure,” as you will see in the applications for triangles presented in
Chapter 5. To find values of trigonometric functions when θ is a multiple
π
of __
or 90º, you can use the above definition and mentally rotate (1, 0).
2
Example 1
Evaluate cos π and sin π.
Solution Because no degree sign is given, π is measured in radians. Think
of Rπ(1, 0), the image of (1, 0) under a rotation of π. Rπ(1, 0) = (–1, 0). So,
by definition, (cos π, sin π) = (–1, 0). Thus, cos π = –1
and sin π = 0.
Check Use a calculator. Make sure it is in radian mode.
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Chapter 4
Because π radians = 180º, Example 1 shows that cos 180º = –1 and
π
sin 180º = 0. Cosines and sines of other multiples of __
or 90º are shown
2
on the unit circle below.
y
(0, 1) = (cos 90˚, sin 90˚)
= cos π2 , sin π2
(-1, 0) = (cos 180˚, sin 180˚)
= (cos π, sin π)
x
(1, 0) = (cos 0˚, sin 0˚)
= (cos 0, sin 0)
(0, -1) = (cos 270˚, sin 270˚)
3π
= cos 3π
2 , sin 2
The third most common circular function is defined in terms of the sine
and cosine functions. The tangent of θ (abbreviated tan θ) equals the
ratio of sin θ to cos θ.
Definition of Tangent
sin θ
For all real numbers θ, provided cos θ ≠ 0, tan θ = _____
.
cos θ
When cos θ does equal zero, which occurs at any odd multiple of 90º, tan
θ is undefi ned.
GUIDED
Example 2
a. Evaluate tan π.
b.
Evaluate tan (–270º).
Solution
a. From Example 1, "#$ π = ? %&' $(& π = ? . *# +%& π = ? .
b. "#$ ,–-./0) = ? , $(& ,–-./0) = ? , $# +%& ,–-./0) ($ ? .
For any value of θ, you can approximate sin θ, cos θ, and tan θ to the
nearest tenth with a good drawing.
Activity
y
MATERIALS compass, protractor, graph paper, and calculator
1
Step 1 Work with a partner. Draw a set of coordinate axes on graph paper.
Let each square on your grid have side length 0.1 unit. With the origin
as center, draw a circle of radius 1. Label the figure as at the right.
Step 2 a. Use a protractor to mark the image of A = (1, 0) under a rotation
of 50º. Label this point P1, as shown at the right.
b. Use the grid to estimate the x- and y-coordinates of P1.
c. Estimate the slope of OP
##$1.
d. Use your calculator to find cos 50º, sin 50º, and tan 50º. Make
sure your calculator is set to degree mode.
230
P1
-1
O
Ax
1
-1
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Lesson 4-2
Step 3 a. Use a protractor to mark the image of A under R155º. Label this
point P 2.
b. Use the grid to estimate the x- and y-coordinates of P 2. Estimate
the slope of OP
!!"2.
c. With your calculator, find cos 155º, sin 155º, and tan 155º.
Step 4 Repeat Step 3 if the rotation has magnitude –100º. Call the image P 3.
Step 5 How is tan θ related to the slope of the line through the origin
and Rθ(A)?
You can find better approximations to other values of sin θ, cos θ, or
tan θ using a calculator.
Example 3
y
B = R(1, 0)
Suppose the tips of the arms of a starfish determine the vertices of a
regular pentagon. The point A = (1, 0) is at the tip of one arm, and so
is one vertex of a regular pentagon ABCDE inscribed in the unit circle, as
shown at the right. Find the coordinates of B to the nearest thousandth.
C
x
A = (1, 0)
D
Solution Since the full circle measures 2π around, the measure of arc
2π
2!
2!
___
___
AB is ___
2π (1, 0). Consequently, B = (cos 5 , sin 5 ). A
5 . So B = R ___
5
E
calculator shows B ≈ (0.309, 0.951).
y
For a given value of θ, you can determine whether
(-, +)
(+, +)
sin θ, cos θ, and tan θ are positive or negative
P
without using a calculator by using coordinate
θ
x
geometry and the unit circle. The cosine is positive
when Rθ(1, 0) is in the first or fourth quadrant. The
sine is positive when the image is in the first or
(+, -)
(-,-)
second quadrant. The tangent is positive when the
sine and cosine have the same sign and negative
P = (cos θ, sin θ) = R!(1, 0)
when they have opposite signs. The following table
summarizes this information for values of θ between 0 and 360º or 2π.
θ (radians)
θ (degrees)
quadrant of Rθ(1, 0)
cos θ
sin θ
tan θ
π
0 < θ < __
0º < θ < 90º
first
+
+
+
90º < θ < 180º
second
–
+
–
180º < θ < 270º
third
–
–
+
270º < θ < 360º
fourth
+
–
–
2
π
__
2
<θ<π
3π
π < θ < ___
2
3π
___
2
< θ < 2π
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Chapter 4
The applications of sines, cosines, and tangents are many and diverse,
including the location of points in the plane and the calculation of certain
distances.
Example 4
As of 2008, the largest Ferris wheel in North America is the Texas Star at Fair
Park in Dallas, Texas. Its seats hang from 44 spokes. This Ferris wheel is 212
feet tall. How high is the seat off the ground as you travel around the wheel?
Solution We need to make some assumptions. Assume that you get on the
Ferris wheel when the seat is at the wheel’s lowest point and
that this is at ground level. Also assume the seat is the same
distance directly below the end of the spoke the entire way
around.
The key to answering the question is to realize that the height
of the seat is determined by the magnitude of rotation of
the spoke from the horizontal. To see this, imagine the Ferris
wheel on a coordinate system whose origin is the center
of the wheel. Think of the circle centered at the origin with
radius 106 feet. By the definition of the sine, when the
spoke has turned θ counterclockwise from the horizontal,
the height of the end of the spoke above the center of the
wheel is given by 106 sin θ.
Add the radius 106 to get the height of the seat above the ground. Thus, in
general, a seat that has been rotated θ counterclockwise from the horizontal is
at a height
106 + 106 sin θ
feet above the ground. Thus, when one seat is at the bottom,
going counterclockwise from the right-most seat, the 44 seats on
the Ferris wheel are at heights
106 + 106 sin 0
(106 cos 90˚, 106 sin 90˚)
= 106 feet
( 2! ) ≈ 121 feet
2!
106 + 106 sin ( 2 • ___
≈ 136 feet
44 )
2!
106 + 106 sin ( 3 • ___
≈ 150 feet
44 )
2!
106 + 106 sin ( 4 • ___
≈ 163 feet
44 )
106 + 106 sin ___
44
(106 cos 180˚, 106 sin 180˚)
(106, 0)
and so on.
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Lesson 4-2
Questions
COVERING THE IDEAS
1. Suppose the point A = (1, 0) is rotated a magnitude θ around the
point O = (0, 0).
a. cos θ is the ? of Rθ(A).
b. sin θ is the ? of Rθ(A).
In 2–4, use the figure at the right. Which point is Rθ(1, 0) for the given
value of θ?
2. 3π
3. –50π
4. –450º
5. How is tan θ related to cos θ and sin θ?
y
B = (0, 1)
x
A = (1, 0)
C = (-1, 0)
In 6–8, give exact values without a calculator.
6. a. sin (–270º)
b. cos (–270º) c. tan (–270º)
7. a. sin 3π
b. cos 3π
c. tan 3π
8. a. sin 0
b. cos 0
c. tan 0
9. a. Give two values of θ in degrees for which tan θ is undefined.
b. Give two values of θ in radians for which tan θ is undefined.
D = (0, -1)
In 10 and 11, find the coordinates of the indicated image to the nearest
thousandth.
10. R67º
11. R1 (radian)
12. a. Use a calculator to approximate tan 200º to three decimal places.
b. Use a picture to explain how you could have found the sign of
tan 200º without using a calculator.
In 13 and 14, let P = Rθ(1, 0).
13. If P is in the fourth quadrant, state the sign of the following.
a. cos θ
b. sin θ
c. tan θ
14. If cos θ < 0 and sin θ < 0, in what quadrant is P?
In 15–17, refer to Example 4.
15. How high is the seat above the ground when it is at the top of the
Ferris wheel?
π
16. How high is the seat above the ground when it has been rotated __
3
from the horizontal?
17. Suppose the seat next to you is at ground level. How high are you
off the ground?
APPLYING THE MATHEMATICS
18. a. In the pentagon of Example 3, find the coordinates of C, D, and
E to the nearest thousandth.
b. Why do you only need to use a calculator for one of the points?
19. Find three values of θ for which cos θ = –1.
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Chapter 4
20. For what values of θ between 0 and 2π is sin θ positive?
21. As θ increases from 0 to 90º, tell whether cos θ increases or
y
decreases.
22. The name “tangent function” is derived from the use of the word
Q
P
“tangent” in geometry. Here is how. At the right, line " is tangent
to the unit circle at A = (1, 0), P is the image of a rotation of A with
""# intersects " at Q.
magnitude θ and center O, and OP
π
__
a. When 0 < θ < 2 , prove that QA = tan θ.
b. Draw a diagram similar to the one at the right for the case of
π
__
< θ < π. Explain how to find tan θ from your diagram.
2
O
x
A = (1, 0)
θ
!
REVIEW
23. Convert __56 revolution clockwise to degrees. (Lesson 4-1)
A
2π
Let A$ be the image of A = (1, 0) under the rotation of – ___
with
3
24.
45˚
center (0, 0). Give two other magnitudes of the rotation with center
(0, 0) such that the image of A is A$. (Lesson 4-1)
1
___
25. In isosceles %ABC at the right, AB = 1. What is the length of BC ?
(Previous Course)
45˚
C
B
____
___
26. %EQU is equilateral, UI ⊥ EQ , and EU = k as shown at the right.
U
a. Find EI in terms of k.
b. Find UI in terms of k. (Previous Course)
k
27. Suppose (x, y) is a point in the first quadrant. Give the coordinates
of its image after each transformation. (Previous Course)
a. reflection over the y-axis
b. reflection over the x-axis
c. rotation of 180º around (0, 0)
E
60˚
I
Q
28. Skill Sequence Simplify in your head. (Previous Course)
√(
5
___
1
__
a.
13
__
7
__
13
b.
13
___
7
__
13
1
__
c.
3
___
√(
5
___
3
EXPLORATION
29. The first Ferris wheel was designed by George Washington Gale
Ferris, Jr., a Pittsburgh bridge builder, for the World’s Columbian
Exposition in Chicago in 1892–1893. It was also the largest Ferris
wheel ever built. It could seat 2160 people at one time. Research
this Ferris wheel for the additional information needed to answer
Questions 15–17. Then answer the questions.
The World’s Columbian
Exposition was a celebration
of the 400th anniversary of
Columbus arriving in the new
world.
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