Transcript Document

13.3 Evaluating Trigonometric
Functions
Evaluating Trigonometric Functions Given a Point
Let (3, – 4) be a point on the
terminal side of an 0 angle in
standard position. Evaluate the
six trigonometric functions
of 0 .
r
SOLUTION
Use the Pythagorean theorem to find the value of r.
r = x2 + y2
= 3 2 + (– 4) 2
= 25
= 5
0
(3, – 4)
Evaluating Trigonometric Functions Given a Point
Using x = 3, y = – 4, and r = 5,
you can write the following:
r
0
(3, – 4)
y
sin 0 = r = – 4
5
x
cos 0 = r =
3
5
y
tan 0 = x = – 4
3
csc 0 = yr = – 5
4
r
sec 0 = x =
5
3
x
cot 0 = y = – 3
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 0 be an angle in standard position. Its reference angle
is the acute angle 0' (read theta prime) formed by the terminal
side of 0 and the x-axis.
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
90° < 0 < 180°;
π < <
π
0
2
0'
Degrees:
0' = 180° – 0
Radians:
0' =
π –0
0
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
180° < 0 < 270°;
3π
π < 0 <
2
0
0'
Degrees:
0' = 0 – 180°
Radians:
0' = 0 – π
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
270° < 0 < 360°;
3π < <
2π
0
2
0
0'
Degrees:
0' = 360° – 0
Radians:
0' = 2π – 0
Finding Reference Angles
Find the reference angle 0' for each angle 0 .
5π
0 = –320°
6
SOLUTION
7π
7π <is3π , the
is
coterminal
with
and
π
<
Because 270°
<
<
360°,
the
reference
angle
0
0
6
6
2
0' = 360° – 320° = 40°.7π
reference angle is 0' =
– π = π .
6
6
Evaluating Trigonometric Functions Given a Point
CONCEPT
SUMMARY
EVALUATING TRIGONOMETRIC FUNCTIONS
Use these steps to evaluate a trigonometric function of
any angle 0.
1
Find the reference angle 0'.
2
Evaluate the trigonometric function for angle 0'.
3
Use the quadrant in which 0 lies to determine the
sign of the trigonometric function value of 0 .
Evaluating Trigonometric Functions Given a Point
CONCEPT
SUMMARY
EVALUATING TRIGONOMETRIC FUNCTIONS
Signs of Function Values
Quadrant II
Quadrant I
sin 0 , csc 0 : +
sin 0 , csc 0 : +
cos 0 , sec 0 : –
cos 0 , sec 0 : +
tan 0 , cot 0 : –
tan 0 , cot 0 : +
Quadrant III
sin 0 , csc 0 : –
Quadrant IV
sin 0 , csc 0 : –
cos 0 , sec 0 : –
cos 0 , sec 0 : +
tan 0 , cot 0 : +
tan 0 , cot 0 : –
Using Reference Angles to Evaluate Trigonometric Functions
Evaluate tan (– 210°).
0' = 30°
SOLUTION
0 = – 210°
The angle – 210° is coterminal with 150°.
The reference angle is 0' = 180° – 150° = 30° .
The tangent function is negative in Quadrant II,
so you can write:
tan (– 210°) = – tan 30° = –
3
3
Using Reference Angles to Evaluate Trigonometric Functions
Evaluate csc 11π .
4
0' =
π
4
SOLUTION
The angle 11π is coterminal with 3π .
4
4
π
3π
The reference angle is 0' = π –
=
.
4
4
The cosecant function is positive in Quadrant II,
so you can write:
csc 11π = csc π = 2
4
4
11π
0= 4