Download Powerpoint

Lesson 4.4
Trigonometric Functions
of
Any Angle
Trigonometric Functions
of Any Angle
Definitions of Trigonometric Functions of Any Angle:
Let  be an angle in standard position with (x, y) a point on the
Terminal side of  and r  x 2  y 2  0
y
sin  
r
x
cos  
r
r
csc   , y  0
y
r
sec   , x  0
x
x
y
tan   , x  0 cot   , x  0
y
x
r

Trigonometric Functions
of Any Angle
Example 1: Let (8, - 6) be a point on the terminal side of . Find the sine,
cosine, and tangent of .
Solution:
 8   6   r 2
2
Step 1: Find r.
2
64  36  r 2
100  r 2
10  r
Step 2: Apply the definitions for sine,
cosine, and tangent.
x 8 4
cos    
r 10 5
y 6
3
tan   

x
8
4
y
6
3
sin      
r
10
5

Trigonometric Functions
of Any Angle
The signs of the trigonometric functions in the four quadrants can be
easily determined by applying CAST. CAST let’s one know where the
trigonometric functions are positive.
y
S
Sine & Cosecant
A
All trig functions are positive.
are positive.
T
x
C
Tangent & Cotangent
Cosine & Secant
are positive.
are positive.
Remember the acronym:
All Students Take Calculus
Trigonometric Functions
of Any Angle
Example 2: Given sin   11 and tan   0, find the value of the remaining
trig functions. 61
Step 1: Determine the quadrant that the terminal side of  lies.
Sine is positive in Quad I and Quad II, while tangent is
positive in Quad I and Quad III. Therefore, the terminal side
must lie in Quad I.
Step 2: Determine the value of r using
the given value of sine.
r  x2  y 2
61  x 2  112
 61 
2

x 2  121
3721  x 2  121
3600  x 2
60  x

2
Step 3: State the values for the
remaining trig functions by
applying the definitions.
60
61
cos  
sec  
61
60
11
60
tan  
cot  
60
11
61
csc  
11
Trigonometric Functions
of Any Angle
The values of trigonometric functions of angles greater than 90 can be
determined by using a reference angle.
Definition of a reference angle:
Let  be an angle in standard position. Its reference angle is the
acute positive angle  ′ formed by the terminal side of  and the
nearest x-axis.
′


′
In Quad II
    
In Quad III
   

′
In Quad IV
   2  
Trigonometric Functions
of Any Angle
Example 3: Find the reference angle for
7
9
Step 1: Determine the quadrant that terminal side lies.
The terminal side for this angle lies in Quad II.
Step 2: Determine the value of the nearest x-axis.
The nearest x-axis holds a value of .
Step 3: Calculate the value for the reference angle. Remember
the reference angle must be an acute angle and positive.
    
   
 
2
9
7
9
Trigonometric Functions
of Any Angle
Example 4: Find the exact values of the six trigonometric functions
for   10
3
First, sketch the angle and determine the angle’s simplest positive
coterminal angle.
10
4
 2 
3
3
Second, determine the new angle’s reference angle based on where
the terminal side lies.
4

 
3
3
′
Third, give the trigonometric values for the original
angle based on the quadrant the terminal side is located
and the reference angle.
10
3
10
2 3

3
2
10
1
cos

3
2
sin
tan
10
 3
3

3
3
10
sec
 2
3
csc
cot
10
3

3
3

Trigonometric Functions
of Any Angle
Try these:
15
17
8
cos   
17
15
tan  
8
sin   
1. Determine the exact values of the six
trigonometric functions of the angle 
given (- 8, - 15) lies on the terminal side.
2. Find the values of the six trigonometric
functions of  giventan  = - 4/3 and sin  < 0.
sin   
cos  
3
5
tan   
3. Find the reference angle for:
   197 180  17
a. 197
12
2
   2 

b. 12/7
c. - 3.68
7
7
4
5
17
15
17
sec   
8
8
cot  
15
csc   
csc   
sec  
4
3
5
4
5
3
cot   
3
4
3.68  6.28  3.00
   3.14  3.00  0.14
Trigonometric Functions
of Any Angle
What you should know:
1. How to evaluate the trigonometric functions of any angle.
2. How find and use the reference angle to evaluate
trigonometric functions.