Download 13 - cloudfront.net

13.3 Evaluate Trigonometric Functions of Any Angle
Goal  Evaluate trigonometric functions of any angle.
Your Notes
VOCABULARY
Unit circle
The circle x2 + y2 = 1, which has center (0, 0) and radius 1
Quadrantal angle
An angle in standard position whose terminal side lies on an axis
Reference angle
The reference angle for  is the acute angle ’ formed by the terminal side of  and the
x-axis.
GENERAL DEFINITIONS OF TRIGONOMETRIC FUNCTIONS
Let  be an angle in standard position, and let (x, y) be the point where the terminal side
of  intersects the circle x2 + y2 = r2. The six trigonometric functions of  are defined as
follows:
sin  =
y
r
csc  =
r
,y  0
y
cos  =
x
r
sec  =
r
,x  0
x
tan  =
y
,x  0
x
cot  =
x
,y  0
y
Your Notes
Example 1
Evaluate trigonometric functions given a point
Let (12, 5) be a point on the terminal side of an angle  in standard position. Evaluate
the six trigonometric functions of .
Use the Pythagorean theorem to find the value of r.
r = x2  y2 =
2
(  12 ) 2  5 =
169 = _13_
Using x = 12, y = 5, and r = _13_ , you can write:
sin  =
5
y
=
13
r
csc  =
cos  =
x
r
=
12
13
sec  =
13
r
=
12
x
tan  =
y
= 
5
cot  =
x
y
x
12
r
y
=
13
5
= 
12
5
Checkpoint Complete the following exercise.
1. Let (6, 8) be a point on the terminal side of an angle  in standard position.
Evaluate the six trigonometric functions of .
sin = 
sec =
4
3
4
5
, cos = ,tan = 
, csc = 
5
5
3
4
5
3
, cot = 
3
4
,
THE UNIT CIRCLE
The circle x2 + y2 = 1, which has center (0, 0) and radius 1, is called the unit circle.
sin  =
y
y
=
=_y_
r
1
cos  =
x
x
=
=_x_
r
1
Your Notes
Example 2
Use the unit circle
Use the unit circle to evaluate the six trigonometric functions of  = 450°.
Draw the unit circle, then draw the angle  = 450° in standard position. The terminal side
of  intersects the unit circle at (_0_, _1_), so use x = _0_ and y = _1_ to evaluate the
trigonometric functions.
sin  =
y 1
= =_1_
1
r
csc  =
cos  =
x
0
= =_0_
r
1
sec  =
tan  =
y
cot  =
x
=
1
_undefined_
0
r
1
=
=_1_
y
1
r
x
= 1 _undefined_
0
x
0
=
=_0_
y
1
Checkpoint Complete the following exercise.
2. Use the unit circle to evaluate the six trigonometric functions of  = 360°.
sin = 0, cos = 1, tan = 0, csc is undefined, sec = 1, cot is undefined.
Your Notes
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°(

2
<  < 2).
Quadrant II
Quadrant III
Quadrant IV
Degrees:
’ =180°  
’ =   180°
’ =360°  
Radians:
’ =  
’ = – 
’ =2  
Example 3
Find reference angles
Find the reference angle ’ for (a)  = 165° and
7
(b)  =
.
4
Solution
a. Note that  is coterminal with _195°_, whose terminal side lies in Quadrant _III_. So,
’ = _195°_  _180°_ = _15°_ .
b. The terminal side of  lies in Quadrant _IV_ . So,
’ = _2_
7

=
.
4
4
Your Notes
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 functions for _’_.
STEP 3 Determine the sign of the trigonometric function value from the
quadrant in which __ lies.
Example 4
Use reference angles to evaluate functions

Evaluate (a) cos(225° ) and (b) cot 10 .
3
a. The angle 225° is coterminal with _135°_. The reference angle is ’ = _180°_ 
_135°_ = _45°_ . The cosine function is negative in Quadrant _II_, so you can
write:
2
cos(225°) = cos( _45°_) = 
2
4

b. The angle 10 is coterminal with
. The reference
3
3
angle is ' =

4
 __=
. The cotangent
3
3
function is positive in Quadrant _III_, so you can write:
cot (

10
) =cot
3
3
=
3
3
Checkpoint Complete the following exercises.
3. Find the reference angle for  = 150°.
30°
4. Evaluate sec 300°.
2
Homework
________________________________________________________________________
________________________________________________________________________