8.3 Trigonometric Functions of Any Angle
8.3
Trigonometric Functions of Any Angle
Essential Question How can you use the unit circle to define the
trigonometric functions of any angle?
Let be an angle in standard position with (x, y) a point on the terminal side of and
r = -- x2 + y2 0. The six trigonometric functions of are defined as shown.
sin = --yr
csc = --yr, y 0
y
(x, y)
cos = --xr tan = --xy, x 0
sec = --xr, x 0 cot = --xy, y 0
r
x
Writing Trigonometric Functions
Work with a partner. Find the sine, cosine, and tangent of the angle in standard position whose terminal side intersects the unit circle at the point (x, y) shown.
( a. -1 , 3 2 2
y
( b. -1 , 1 22
y
c.
y
(
(
x
d.
y
e.
x
x
(0, -1)
y
f.
y
(-1, 0)
x
x
x
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.
(
(
(1,- 3 22
( 1 , -1 22
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
c. secant
d. cosecant
Section 8.3 Trigonometric Functions of Any Angle 425
8.3 Lesson
Core Vocabulary
unit circle, p. 427 quadrantal angle, p. 427 reference angle, p. 428 Previous circle radius Pythagorean Theorem
What You Will Learn
Evaluate trigonometric functions of any angle. Find and use reference angles to evaluate trigonometric functions.
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 be an angle in standard position, and let (x, y)
y
be the point where the terminal side of intersects
the circle x2 + y2 = r2. The six trigonometric
functions of are defined as shown.
(x, y)
sin = --yr
csc = --yr, y 0
r
x
cos = --xr
sec = --xr, x 0
tan = --xy, x 0
cot = --yx, y 0
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 in standard position. Evaluate the six trigonometric functions of .
SOLUTION
y
(-4, 3)
r
Use the Pythagorean Theorem to find the length of r.
x
r = -- x2 + y2
= -- (-4)2 + 32
--
= 25
= 5
Using x = -4, y = 3, and r = 5, the values of the six trigonometric functions of are:
sin = --yr = --53
csc = --yr = --35
cos = --xr = ---54
sec = --xr = ---45
tan = --yx = ---43
cot = --xy = ---34
426 Chapter 8 Trigonometric Ratios and Functions
ANOTHER WAY
The general circle x2 + y2 = r2 can also be used to find the six trigonometric functions of . The terminal side of intersects the circle at (0, -r). So,
sin = --yr = -- -rr = -1.
The other functions can be evaluated similarly.
Core Concept
The Unit Circle
y
The circle x2 + y2 = 1, which has center (0, 0)
and radius 1, is called the unit circle. The values
of sin and cos are simply the y-coordinate and
x-coordinate, respectively, of the point where the
terminal side of intersects the unit circle.
x
sin = --yr = --1y = y
r = 1 (x, y)
cos = --xr = --1x = x
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 an axis. The measure of a quadrantal angle is always a multiple of 90?, or --2 radians.
Using the Unit Circle
Use the unit circle to evaluate the six trigonometric functions of = 270?.
SOLUTION
y
Step 1 Draw a unit circle with the angle = 270? in
standard position.
Step 2 Identify the point where the terminal side
of intersects the unit circle. The terminal
x
side of intersects the unit circle at (0, -1).
Step 3 Find the values of the six trigonometric functions. Let x = 0 and y = -1 to evaluate the trigonometric functions.
(0, -1)
sin = --yr = -- -11 = -1
csc = --yr = -- -11 = -1
cos = --xr = --10 = 0
sec = --xr = --01 undefined
tan = --yx = -- -01 undefined
cot = --xy = -- -01 = 0
Monitoring Progress
Help in English and Spanish at
Evaluate the six trigonometric functions of .
1.
y
2. (-8, 15) y
3.
y
x
x
x
(3, -3)
(-5, -12)
4. Use the unit circle to evaluate the six trigonometric functions of = 180?.
Section 8.3 Trigonometric Functions of Any Angle 427
READING
The symbol is read as "theta prime."
Reference Angles
Core Concept
Reference Angle Relationships
Let be an angle in standard position. The reference angle for is the acute angle formed by the terminal side of and the x-axis. The relationship between and is shown below for nonquadrantal angles such that 90? < < 360? or, in radians, --2 < < 2.
Quadrant II
Quadrant III
Quadrant IV
y
y
y
x
x
x
Degrees: = 180? - Degrees: = - 180? Degrees: = 360? -
Radians: = -
Radians: = -
Radians: = 2 -
Finding Reference Angles
Find the reference angle for (a) = -- 53 and (b) = -130?.
SOLUTION
a. The terminal side of lies in Quadrant IV. So,
y
= 2 - -- 53 = --3 . The figure at the right shows
x
= -- 53 and = --3 .
b. Note that is coterminal with 230?, whose terminal side
lies in Quadrant III. So, = 230? - 180? = 50?. The
figure at the left shows = -130? and = 50?.
y
x
Reference angles allow you to evaluate a trigonometric function for any angle . The sign of the trigonometric function value depends on the quadrant in which lies.
Core Concept
Evaluating Trigonometric Functions
Use these steps to evaluate a trigonometric function for any angle :
Step 1 Find the reference angle .
Step 2 Evaluate the trigonometric function for .
Step 3
Determine the sign of the trigonometric function value from the quadrant in which lies.
Signs of Function Values
Quadrant II sin , csc : + cos , sec : - tan , cot : -
y Quadrant I sin , csc : + cos , sec : + tan , cot : +
Quadrant III sin , csc : -
cos , sec : -
tan , cot : +
Quadrant IV x sin , csc : -
cos , sec : +
tan , cot : -
428 Chapter 8 Trigonometric Ratios and Functions
INTERPRETING MODELS
This model neglects air resistance and assumes that the projectile's starting and ending heights are the same.
Using Reference Angles to Evaluate Functions Evaluate (a) tan(-240?) and (b) csc -- 176.
SOLUTION
a. The angle -240? is coterminal with 120?. The reference angle is = 180? - 120? = 60?. The tangent function is negative in Quadrant II, so
--
tan(-240?) = -tan 60? = -3.
b. The angle -- 176 is coterminal with -- 56. The reference angle is = - -- 56 = --6 . The cosecant function is positive in Quadrant II, so csc -- 176 = csc --6 = 2.
y
= 60?
x
= -240?
=
6
y
x
=176
Solving a Real-Life Problem
The horizontal distance d (in feet) traveled by a projectile launched at an angle and with an initial speed v (in feet per second) is given by
d = -- 3v22 sin 2.
Model for horizontal distance
Estimate the horizontal distance traveled by a golf ball
that is hit at an angle of 50? with an initial speed of
105 feet per second.
50?
SOLUTION
Note that the golf ball is launched at an angle of = 50? with initial speed of v = 105 feet per second.
d = -- 3v22 sin 2
= -- 130252 sin(2 50?)
Write model for horizontal distance. Substitute 105 for v and 50? for .
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.
5. 210?
6. -260?
7. -- -97
Evaluate the function without using a calculator.
8. -- 154
9. cos(-210?)
10. sec -- 114
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? and with an initial speed of 27 feet per second.
Section 8.3 Trigonometric Functions of Any Angle 429
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