1. Trigonometric Functions of Any Angle
Skill 28

2. Objectives… Evaluate trigonometric functions of any angle
Find reference angles
Evaluate trigonometric functions of real numbers

4. Example–Evaluating Trigonometric Functions
Let (–3, 4) be a point on the terminal side of . Find the sine, cosine, and tangent of .
x = –3, y = 4

5. Example–Solution

6. The signs of the trigonometric functions in the four quadrants can be determined easily from the definitions of the functions.

7. Reference Angles
The values of the trigonometric functions of angles greater than 90  (or less than 0 ) can be determined from their values at corresponding acute angles called reference angles.

8. Reference Angles
Reference angles for  in Quadrants II, III, and IV.

9. Example–Finding Reference Angles
Find the reference angle  .
a.  = 300  c.  = –135 
Solution:
a. Because 300  lies in Quadrant IV, the angle it makes with the x-axis is
  = 360  – 300 
= 60 .

10. Example–Solution b.  = 2.3  /2  1.5708   3.1416
Quadrant II, Reference angle
  =  – 2.3 

11. Example–Solution c.  = –135  Coterminal with 225 
Quadrant III, Reference angle
  = 225  – 180 
= 45 .

12. Trigonometric Functions of Real Numbers
To see how a reference angle is used to evaluate a trigonometric function, consider the point (x, y) on the terminal side of .

13. Trigonometric Functions of Real Numbers

14. Cosine is negative in Quadrant III
Example–Trigonometric Functions of Nonacute Angles
 = 4 /3… Quadrant III
Reference angle…
  = (4 /3) – 
  =  /3
Cosine is negative in Quadrant III

15. tan (–210 ) –210  + 360  = 150  –210  is coterminal with 150 .
Example–Trigonometric Functions of Nonacute Angles
tan (–210 )
–210   = 150 
–210  is coterminal with 150 .
Reference angle
  = 180  – 150 
  = 30 
Tangent is negative in Quadrant II.

16. 11 /4 is coterminal with 3/4. Reference angle   =  – (3 /4)
Example–Trigonometric Functions of Nonacute Angles
(11 /4) – 2 = 3 /4,
11 /4 is coterminal with 3/4.
Reference angle
  =  – (3 /4)
  =  /4
Cosecant is positive in Quadrant II

17. 28: Trigonometric Functions of Any Angle
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