Trigonometric Functions of Any
Angle
4.4
Definitions of Trigonometric Functions of Any Angle
• Let is be any angle in standard position, and let P = (x, y) be a point on the terminal side of . If r = x2 + y2 is the distance from (0, 0) to (x, y), the six trigonometric functions of are defined by the following ratios.
sin y
r, cos x
r, tan y
x, x 0
csc r
y,y 0 sec
r
x, x 0 cot
x
y,y 0
Let P = (-3, -4) be a point on the terminal side of . Find each of the six trigonometric functions of .
Solution The situation is shown below. We need values for x, y, and r to evaluate all six trigonometric functions. We are given the values of x and y. Because P = (-3, -4) is a point on the terminal side of , x = -3 and y = -4. Furthermore,
r
x = -3 y = -4
P = (-3, -4)
x
y
-5
5
-5
5
Example
222 ryx
The bottom row shows the reciprocals of the row above.The bottom row shows the
reciprocals of the row above.
sin yr
45
45
, cos xr
35
35
, tan yx
4 3
43
csc r
y
5
4
5
4, sec
r
x
5
3
5
3, cot
x
y
3
4
3
4
Example Cont.
SolutionNow that we know x, y, and r, we can find the
six trigonometric functions of .
ExampleLet tan θ = -2/3 and cos θ > 0. Find each of the six trigonometric functions of .
x
y
3
2tan
222 ryx 222 )3()2( r
r13
13
133
13
3cos
We have to be in Quadrant IV
2
3cot
3
13sec
13
132
13
2sin
2
13csc
x
y
Quadrant IISine and cosecant positive
Quadrant IAll functions
positive
Quadrant IIItangent and cotangent positive
Quadrant IVcosine and
secant positive
The Signs of the Trigonometric Functions
All Students Take Calculus
Definition of a Reference Angle
• Let be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle ´ prime formed by the terminal side or and the x-axis.
Example
a
b
a
b
P(a, b)
Find the reference angle , for the following angle: =315º
Solution:
=360º - 315º = 45º
Example
Find the reference angles for:345
135
6
5
4
11
15345360
45180225360135
66
5
6
6
6
5
44
3
4
32
4
11
Using Reference Angles to Evaluate Trigonometric Functions
• The values of a trigonometric functions of a given angle, , are the same as the values for the trigonometric functions of the reference angle, ´, except possibly for the sign. A function value of the acute angle, ´, is always positive. However, the same functions value for may be positive or negative.
A Procedure for Using Reference Angles to Evaluate Trigonometric Functions
• The value of a trigonometric function of any angle is found as follows:
• Find the associated reference angle, ´, and the function value for ´.
• Use the quadrant in which lies to prefix the appropriate sign to the function value in step 1.
Use reference angles to find the exact value of the following trigonometric functions.
Solution
a. We use our two-step procedure to find sin 135°.Step 1 Find the reference angle, ´, and sin ´.
135º terminates in quadrant II with a reference angle ´ = 180º – 135º = 45º.
x
y
135°45°
a. sin 135°
Example
Solution
The function value for the reference angle is sin 45º = 2 / 2.
Step 2 Use the quadrant in which è lies to prefix the appropriate sign to the function value in step 1. The angle 135º lies in quadrant II. Because the sine is positive in quadrant II, we put a + sign before the function value of the reference angle. Thus, sin135= +sin45=2 / 2
Example cont.
Example
• Evaluate:
3
4cos
3cot
2
1
3cos
3
4cos
3
3
3
1
3tan
1
3cot
3cot
Top Related