4.4 Trigonometric Functions of Any Angle

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4.4

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Trigonometry

Trigonometric Functions of Any Angle

What you should learn

? Evaluate trigonometric

functions of any angle.

? Use reference angles

to evaluate trigonometric

functions.

? Evaluate trigonometric

functions of real numbers.

Why you should learn it

You can use trigonometric

functions to model and solve

real-life problems. For instance,

in Exercise 87 on page 319, you

can use trigonometric functions

to model the monthly normal

temperatures in New York City

and Fairbanks, Alaska.

Introduction

In Section 4.3, the definitions of trigonometric functions were restricted to acute

angles. In this section, the definitions are extended to cover any angle. If
is an

acute angle, these definitions coincide with those given in the preceding section.

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 
x2  y2  0.

sin


y

r

y

tan
 ,

x

cos


x

r

x0

x

cot
 ,

y

x0

r

csc
 ,

y

r

sec
 ,

x

y

(x , y)

y0

r

y0

¦È

x

Because r 
x 2  y 2 cannot be zero, it follows that the sine and cosine

functions are defined for any real value of
. However, if x  0, the tangent and

secant of
are undefined. For example, the tangent of 90 is undefined.

Similarly, if y  0, the cotangent and cosecant of
are undefined.

Example 1

Evaluating Trigonometric Functions

Let 3, 4 be a point on the terminal side of
. Find the sine, cosine, and

tangent of
.

Solution

Referring to Figure 4.36, you can see that x  3, y  4, and

r 
x 2  y 2 
3 2  42 
25  5.

James Urbach/SuperStock

y

(?3, 4)

So, you have the following.

sin


y 4



r

5

cos


x

3



r

5

tan


y

4



x

3

4

3

r

2

1

?3

FIGURE

?2

4.36

?1

¦È

x

1

Now try Exercise 1.

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Section 4.4

y

¦Ð 0

x

x ................
................

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