1. 13.3 Evaluating Trigonometric Functions

2. terminal side of an angle in standard position. Evaluate the
Evaluating Trigonometric Functions Given a Point
Let (3, – 4) be a point on the
terminal side of an angle in
standard position. Evaluate the
six trigonometric functions
of . r
(3, – 4)
SOLUTION
Use the Pythagorean theorem to find the value of r.
r = x 2 + y 2
= (– 4 ) 2
= 25
= 5

3. you can write the following:
Evaluating Trigonometric Functions Given a Point r
Using x = 3, y = – 4, and r = 5,
you can write the following:
(3, – 4) 4 5
sin = = – y r
csc = = – r y 5 4
cos = = x r 3 5
sec = = r x 5 3
tan = = – y x 4 3
cot = = – x y 3 4

4. ' TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
The values of trigonometric functions of angles greater than
90° (or less than 0°) can be found using corresponding acute angles called reference angles.
Let be an angle in standard position. Its reference angle is the acute angle (read theta prime) formed by the terminal side of and the x-axis. '

5. ' ' ' π TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 90° < < 180°;
90° < < 180°; π 2
< < '
= 180°
Degrees: ' –
Radians: = π – '

6. ' ' ' 3π π π TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 180° < < 270°;
180° < < 270°; π 3π 2
< < ' –
180° =
Degrees: '
Radians: = – π '

7. ' ' ' 3π 2π TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 270° < < 360°;
270° < < 360°; 2π 3π 2
< < ' –
360° =
Degrees: '
Radians: = – 2π '

8. ' ' ' Find the reference angle for each angle . = – 5π 6 = 320°
Finding Reference Angles
Find the reference angle for each angle . '
= – 5π 6
= 320°
SOLUTION
Because is coterminal with and π < < , the
reference angle is = – π = 7π 6 π 3π 2 '
Because 270° < < 360°, the reference angle is
= 360° – 320° = 40°. '

9. ' ' Use these steps to evaluate a trigonometric function of
Evaluating Trigonometric Functions Given a Point
CONCEPT
SUMMARY
EVALUATING TRIGONOMETRIC FUNCTIONS
Use these steps to evaluate a trigonometric function of
any angle .
Find the reference angle . ' 1 2
Evaluate the trigonometric function for angle . ' 3
Use the quadrant in which lies to determine the sign of the trigonometric function value of .

10. Signs of Function Values
Evaluating Trigonometric Functions Given a Point
CONCEPT
SUMMARY
EVALUATING TRIGONOMETRIC FUNCTIONS
Signs of Function Values
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
sin , csc : +
sin , csc : +
cos , sec : –
cos , sec : +
tan , cot : –
tan , cot : +
sin , csc : –
sin , csc : –
cos , sec : –
cos , sec : +
tan , cot : +
tan , cot : –

11. ' ' Evaluate tan (– 210°). SOLUTION
Using Reference Angles to Evaluate Trigonometric Functions
Evaluate tan (– 210°). '
= 30°
SOLUTION
= – 210°
The angle – 210° is coterminal with 150°.
The reference angle is = 180° – 150° = 30° . '
The tangent function is negative in Quadrant II,
so you can write:
tan (– 210°) = – tan 30° = – 3

12. ' ' Evaluate csc . csc = csc = 11π 4 π SOLUTION
Using Reference Angles to Evaluate Trigonometric Functions =
11π 4
Evaluate csc
11π 4 ' = π 4
SOLUTION
The angle is coterminal with
11π 4 3π
The reference angle is = π – = 3π 4 π '
The cosecant function is positive in Quadrant II,
so you can write: 2
csc = csc =
11π 4 π