8.3 Trigonometric Functions of Any Angle
Trigonometric Functions of
Any Angle
8.3
Essential Question
How can you use the unit circle to define the
trigonometric functions of any angle?
Let ¦È be an angle in standard position with (x, y) a point on the terminal side of ¦È and
¡ª
r = ¡Ì x2 + y2 ¡Ù 0. The six trigonometric functions of ¦È are defined as shown.
y
sin ¦È = ¡ª
r
r
csc ¦È = ¡ª, y ¡Ù 0
y
x
cos ¦È = ¡ª
r
r
sec ¦È = ¡ª, x ¡Ù 0
x
y
tan ¦È = ¡ª, x ¡Ù 0
x
x
cot ¦È = ¡ª, y ¡Ù 0
y
y
(x, y)
r
¦È
x
Writing Trigonometric Functions
Work with a partner. Find the sine, cosine, and tangent of the angle ¦È in standard
position whose terminal side intersects the unit circle at the point (x, y) shown.
a.
(
?1 , 3
2 2
(
y
b.
(?12 , 12 (
c.
y
x
y
x
x
(0, ?1)
d.
e.
y
f.
y
y
(?1, 0)
x
x
x
( 12 , ? 2 3 (
( 12 , ?12 (
CONSTRUCTING
VIABLE ARGUMENTS
To be proficient in
math, you need to
understand and use stated
assumptions, definitions,
and previously established
results.
Communicate Your Answer
2. How can you use the unit circle to define the trigonometric functions of any angle?
3. For which angles are each function undefined? Explain your reasoning.
a. tangent
b. cotangent
Section 8.3
Int_Math3_PE_08.03.indd 425
c. secant
d. cosecant
Trigonometric Functions of Any Angle
425
1/30/15 4:48 PM
8.3 Lesson
What You Will Learn
Evaluate trigonometric functions of any angle.
Find and use reference angles to evaluate trigonometric functions.
Core Vocabul
Vocabulary
larry
unit circle, p. 427
quadrantal angle, p. 427
reference angle, p. 428
Previous
circle
radius
Pythagorean Theorem
Trigonometric Functions of Any Angle
You can generalize the right-triangle definitions of trigonometric functions so that they
apply to any angle in standard position.
Core Concept
General Definitions of Trigonometric Functions
Let ¦È be an angle in standard position, and let (x, y)
be the point where the terminal side of ¦È intersects
the circle x2 + y2 = r2. The six trigonometric
functions of ¦È are defined as shown.
y
sin ¦È = ¡ª
r
x
cos ¦È = ¡ª
r
y
tan ¦È = ¡ª, x ¡Ù 0
x
y
¦È
(x, y)
r
r
csc ¦È = ¡ª, y ¡Ù 0
y
r
sec ¦È = ¡ª, x ¡Ù 0
x
x
cot ¦È = ¡ª, y ¡Ù 0
y
x
These functions are sometimes called circular functions.
Evaluating Trigonometric Functions Given a Point
Let (?4, 3) be a point on the terminal side of
an angle ¦È in standard position. Evaluate the
six trigonometric functions of ¦È.
y
¦È
(?4, 3)
SOLUTION
r
Use the Pythagorean Theorem to find the length of r.
x
¡ª
r = ¡Ì x2 + y2
¡ª
= ¡Ì (?4)2 + 32
¡ª
= ¡Ì 25
=5
Using x = ?4, y = 3, and r = 5, the values of the six trigonometric functions of ¦È are:
426
Chapter 8
Int_Math3_PE_08.03.indd 426
y 3
sin ¦È = ¡ª = ¡ª
r 5
r 5
csc ¦È = ¡ª = ¡ª
y 3
4
x
cos ¦È = ¡ª = ?¡ª
r
5
5
r
sec ¦È = ¡ª = ?¡ª
x
4
3
y
tan ¦È = ¡ª = ?¡ª
x
4
4
x
cot ¦È = ¡ª = ?¡ª
y
3
Trigonometric Ratios and Functions
1/30/15 4:48 PM
Core Concept
The Unit Circle
y
The circle + = 1, which has center (0, 0)
and radius 1, is called the unit circle. The values
of sin ¦È and cos ¦È are simply the y-coordinate and
x-coordinate, respectively, of the point where the
terminal side of ¦È intersects the unit circle.
x2
y ?r
sin ¦È = ¡ª = ¡ª = ?1.
r
r
The other functions can be
evaluated similarly.
¦È
x
r=1
y y
sin ¦È = ¡ª = ¡ª = y
r 1
x x
cos ¦È = ¡ª = ¡ª = x
r 1
ANOTHER WAY
The general circle
x2 + y2 = r2 can also be used
to find the six trigonometric
functions of ¦È. The terminal
side of ¦È intersects the circle
at (0, ?r). So,
y2
(x, y)
It is convenient to use the unit circle to find trigonometric functions of quadrantal
angles. A quadrantal angle is an angle in standard position whose terminal side lies on
¦Ð
an axis. The measure of a quadrantal angle is always a multiple of 90?, or ¡ª radians.
2
Using the Unit Circle
Use the unit circle to evaluate the six trigonometric functions of ¦È = 270?.
SOLUTION
y
Step 1 Draw a unit circle with the angle ¦È = 270? in
standard position.
¦È
Step 2 Identify the point where the terminal side
of ¦È intersects the unit circle. The terminal
side of ¦È intersects the unit circle at (0, ?1).
x
Step 3 Find the values of the six trigonometric
functions. Let x = 0 and y = ?1 to evaluate
the trigonometric functions.
(0, ?1)
y ?1
sin ¦È = ¡ª = ¡ª = ?1
r
1
1
r
csc ¦È = ¡ª = ¡ª = ?1
y ?1
x 0
cos ¦È = ¡ª = ¡ª = 0
r 1
r 1
sec ¦È = ¡ª = ¡ª
x 0
y ?1
tan ¦È = ¡ª = ¡ª
x
0
0
x
cot ¦È = ¡ª = ¡ª = 0
y ?1
undefined
Monitoring Progress
undefined
Help in English and Spanish at
Evaluate the six trigonometric functions of ¦È.
1.
2.
y
(?8, 15)
3.
y
¦È
¦È
x
(3, ?3)
y
¦È
x
x
(?5, ?12)
4. Use the unit circle to evaluate the six trigonometric functions of ¦È = 180?.
Section 8.3
Int_Math3_PE_08.03.indd 427
Trigonometric Functions of Any Angle
427
1/30/15 4:48 PM
Reference Angles
Core Concept
READING
Reference Angle Relationships
The symbol ¦È¡ä is read as
¡°theta prime.¡±
Let ¦È be an angle in standard position. The reference angle for ¦È is the acute
angle ¦È¡ä formed by the terminal side of ¦È and the x-axis. The relationship between
¦È and ¦È¡ä is shown below for nonquadrantal angles ¦È such that 90¡ã < ¦È < 360¡ã or,
¦Ð
in radians, ¡ª < ¦È < 2¦Ð.
2
y
¦È¡ä
Quadrant IV
Quadrant III
Quadrant II
y
y
¦È
¦È
¦È
x
Degrees: ¦È ¡ä = 180¡ã ? ¦È
Radians: ¦È ¡ä = ¦Ð ? ¦È
¦È¡ä
x
Degrees: ¦È ¡ä = ¦È ? 180¡ã
Radians: ¦È ¡ä = ¦È ? ¦Ð
¦È¡ä
x
Degrees: ¦È ¡ä = 360¡ã ? ¦È
Radians: ¦È ¡ä = 2¦Ð ? ¦È
Finding Reference Angles
5¦Ð
Find the reference angle ¦È ¡ä for (a) ¦È = ¡ª and (b) ¦È = ?130?.
3
SOLUTION
a. The terminal side of ¦È lies in Quadrant IV. So,
y
x
¦È¡ä
¦È
y
5¦Ð ¦Ð
¦È¡ä = 2¦Ð ? ¡ª = ¡ª. The figure at the right shows
3
3
¦Ð
5¦Ð
¦È = ¡ª and ¦È¡ä = ¡ª.
3
3
b. Note that ¦È is coterminal with 230?, whose terminal side
lies in Quadrant III. So, ¦È¡ä = 230? ? 180? = 50?. The
figure at the left shows ¦È = ?130? and ¦È¡ä = 50?.
¦È
x
¦È¡ä
Reference angles allow you to evaluate a trigonometric function for any angle ¦È. The
sign of the trigonometric function value depends on the quadrant in which ¦È lies.
Core Concept
Evaluating Trigonometric Functions
Use these steps to evaluate a
trigonometric function for any angle ¦È:
Step 1 Find the reference angle ¦È¡ä.
Step 2 Evaluate the trigonometric
function for ¦È¡ä.
Step 3 Determine the sign of the
trigonometric function value
from the quadrant in which
¦È lies.
428
Chapter 8
Int_Math3_PE_08.03.indd 428
Signs of Function Values
Quadrant II
sin ¦È, csc ¦È : +
cos ¦È , sec ¦È : ?
tan ¦È , cot ¦È : ?
Quadrant III
sin ¦È, csc ¦È : ?
cos ¦È , sec ¦È : ?
tan ¦È , cot ¦È : +
Quadrant I
sin ¦È, csc ¦È : +
cos ¦È , sec ¦È : +
tan ¦È , cot ¦È : +
y
Quadrant IV x
sin ¦È, csc ¦È : ?
cos ¦È , sec ¦È : +
tan ¦È , cot ¦È : ?
Trigonometric Ratios and Functions
1/30/15 4:49 PM
Using Reference Angles to Evaluate Functions
17¦Ð
Evaluate (a) tan(?240?) and (b) csc ¡ª.
6
SOLUTION
y
a. The angle ?240? is coterminal with 120?. The reference
angle is ¦È¡ä = 180? ? 120? = 60?. The tangent function ¦È¡ä = 60¡ã
is negative in Quadrant II, so
¡ª
x
tan(?240?) = ?tan 60? = ?¡Ì 3 .
¦È = ?240¡ã
5¦Ð
17¦Ð
b. The angle ¡ª is coterminal with ¡ª. The
6
6
reference angle is
y
5¦Ð ¦Ð
¦È¡ä = ¦Ð ? ¡ª = ¡ª.
6
6
The cosecant function is positive in Quadrant II, so
INTERPRETING
MODELS
This model neglects air
resistance and assumes
that the projectile¡¯s
starting and ending
heights are the same.
¦È¡ä= ¦Ð6
17¦Ð
¦È=
6
17¦Ð
¦Ð
csc ¡ª = csc ¡ª = 2.
6
6
x
Solving a Real-Life Problem
The horizontal distance d (in feet) traveled by a projectile launched at
an angle ¦È and with an initial speed v (in feet per second) is given by
v2
d = ¡ª sin 2¦È.
Model for horizontal distance
32
Estimate the horizontal distance traveled by a golf ball
that is hit at an angle of 50¡ã with an initial speed of
105 feet per second.
50¡ã
SOLUTION
Note that the golf ball is launched at an angle of ¦È = 50? with initial speed
of v = 105 feet per second.
v2
d = ¡ª sin 2¦È
32
1052
= ¡ª sin(2 50¡ã)
32
Write model for horizontal distance.
?
Substitute 105 for v and 50? for ¦È.
¡Ö 339
Use a calculator.
The golf ball travels a horizontal distance of about 339 feet.
Monitoring Progress
Help in English and Spanish at
Sketch the angle. Then find its reference angle.
?7¦Ð
9
Evaluate the function without using a calculator.
6. ?260¡ã
5. 210¡ã
7. ¡ª
15¦Ð
4
8. ¡ª
11¦Ð
4
11. Use the model given in Example 5 to estimate the horizontal distance traveled
by a track and field long jumper who jumps at an angle of 20¡ã and with an initial
speed of 27 feet per second.
9. cos(?210?)
Section 8.3
Int_Math3_PE_08.03.indd 429
10. sec ¡ª
Trigonometric Functions of Any Angle
429
1/30/15 4:49 PM
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