Section 4.4 Trigonometric Functions of Any Angle 537

Section 4.4 Trigonometric Functions of Any Angle

SECTION 4.4

Trigonometric Functions of Any Angle

Cycles govern many aspects of

life��heartbeats, sleep patterns,

seasons, and tides all follow

regular,

predictable

cycles.

Because of their periodic nature,

trigonometric

functions

are

used to model phenomena that

occur in cycles. It is helpful to

apply these models regardless of

whether we think of the domains

of trigonometric functions as sets

of real numbers or sets of angles.

In order to understand and use

models for cyclic phenomena from

an angle perspective, we need to

move beyond right triangles.

Objectives

??? Use the de?nitions of

???

???

???

537

trigonometric functions

of any angle.

Use the signs of the

trigonometric functions.

Find reference angles.

Use reference angles to

evaluate trigonometric

functions.

Trigonometric Functions

of Any Angle

???

In the last section, we evaluated

trigonometric functions of acute

angles, such as that shown in Figure 4.41(a). Note that this angle is in standard

position. The point P = (x, y) is a point r units from the origin on the terminal

side of u. A right triangle is formed by drawing a line segment from P = (x, y)

perpendicular to the x@axis. Note that y is the length of the side opposite u and x is

the length of the side adjacent to u.

Use the de?nitions of

trigonometric functions

of any angle.

y

y

y

u

P  (x, y)

P  (x, y)

y

(a) u lies in

quadrant I.

r

u

u

x

y

r

y

x

x

(b) u lies in

quadrant II.

u

x

x

r

y

x

x

r

y

P  (x, y)

x

P  (x, y)

(c) u lies in

quadrant III.

(d) u lies in

quadrant IV.

FIGURE 4.41

Figures 4.41(b), (c), and (d) show angles in standard position, but they are not

acute. We can extend our de?nitions of the six trigonometric functions to include

such angles, as well as quadrantal angles. (Recall that a quadrantal angle has its

terminal side on the x@axis or y@axis; such angles are not shown in Figure 4.41.) The

point P = (x, y) may be any point on the terminal side of the angle u other than the

origin, (0, 0).

538

Chapter 4 Trigonometric Functions

y

De?nitions of Trigonometric Functions of Any Angle

P  (x, y)

r

y

u

x

x

FIGURE 4.41(a) u lies in quadrant I.

(repeated)

Let u be any angle in standard position and let P = (x, y) be a point on the

terminal side of u. If r = 2x2 + y 2 is the distance from (0, 0) to (x, y), as shown

in Figure 4.41 on the previous page, the six trigonometric functions of U are

de?ned by the following ratios:

y

r

sin u=

csc u= , y  0

r

y

x

r

cos u=

sec u= , x  0

r

x

x

y

tan u= , x  0

cot u= , y  0.

y

x

GREAT QUESTION!

The ratios in the second column

are the reciprocals of the

corresponding ratios in the

first column.

Is there a way to make it

easier for me to remember the

de?nitions of trigonometric

functions of any angle?

Yes. If u is acute, we have the right

triangle shown in Figure 4.41(a).

In this situation, the de?nitions

in the box are the right triangle

de?nitions of the trigonometric

functions. This should make it

easier for you to remember the

six de?nitions.

Because the point P = (x, y) is any point on the terminal side of u other than

the origin, (0, 0), r = 2x2 + y 2 cannot be zero. Examine the six trigonometric

functions de?ned above. Note that the denominator of the sine and cosine functions

is r. Because r ? 0, the sine and cosine functions are de?ned for any angle u. This

is not true for the other four trigonometric functions. Note that the denominator of

y

r

the tangent and secant functions is x: tan u = and sec u = . These functions are

x

x

not de?ned if x = 0. If the point P = (x, y) is on the y@axis, then x = 0. Thus, the

tangent and secant functions are unde?ned for all quadrantal angles with terminal

sides on the positive or negative y@axis. Likewise, if P = (x, y) is on the x@axis, then

x

y = 0, and the cotangent and cosecant functions are unde?ned: cot u = and

y

r

csc u = . The cotangent and cosecant functions are unde?ned for all quadrantal

y

angles with terminal sides on the positive or negative x@axis.

EXAMPLE 1

Evaluating Trigonometric Functions

Let P = (-3, -5) be a point on the terminal side of u. Find each of the six

trigonometric functions of u.

y

5

SOLUTION

u

?5

5

x

The situation is shown in Figure 4.42. 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, -5)

is a point on the terminal side of u, x = -3 and y = -5. Furthermore,

r = 2x2 + y 2 = 2(-3)2 + (-5)2 = 29 + 25 = 234.

r

Now that we know x, y, and r, we can ?nd the six trigonometric functions of u.

Where appropriate, we will rationalize denominators.

?5

P = (?3, ? 5)

x = ?3

FIGURE 4.42

y = ?5

sin u =

y

-5

5 # 234

5234

=

= = r

34

234

234 234

-3

x

3 # 234

3234

=

= = r

34

234

234 234

y

-5

5

tan u = =

=

x

-3

3

cos u =

csc u =

234

r

234

=

= y

-5

5

234

r

234

=

= x

-3

3

-3

x

3

cot u = =

=

�� �� ��

y

-5

5

sec u =

Check Point 1 Let P = (1, -3) be a point on the terminal side of u. Find each

of the six trigonometric functions of u.

Section 4.4 Trigonometric Functions of Any Angle

539

How do we ?nd the values of the trigonometric functions for a quadrantal angle?

First, draw the angle in standard position. Second, choose a point P on the angle��s

terminal side. The trigonometric function values of u depend only on the size of u and

not on the distance of point P from the origin. Thus, we will choose a point that is 1 unit

from the origin. Finally, apply the de?nitions of the appropriate trigonometric functions.

EXAMPLE 2

Trigonometric Functions of Quadrantal Angles

Evaluate, if possible, the sine function and the tangent function at the following

four quadrantal angles:

p

3p

a. u = 0? = 0 b. u = 90? =

c. u = 180? = p d. u = 270? =

.

2

2

SOLUTION

x=1

y

u = 0?

y=0

P = (1, 0)

?1

x

1

r=1

FIGURE 4.43

y

x=0

y=1

P = (0, 1)

1

r=1

u = 90?

?1

x

1

FIGURE 4.44

y

x = ?1

y=0

1

u = 180?

P = (?1, 0)

?1

1

x

r=1

FIGURE 4.45

y

u = 270?

?1

r=1

x

1

?1

P = (0, ?1)

x=0

FIGURE 4.46

y = ?1

a. If u = 0? = 0 radians, then the terminal side of the angle is on the positive

x@axis. Let us select the point P = (1, 0) with x = 1 and y = 0. This point

is 1 unit from the origin, so r = 1. Figure 4.43 shows values of x, y, and r

corresponding to u = 0? or 0 radians. Now that we know x, y, and r, we can

apply the de?nitions of the sine and tangent functions.

y

0

sin 0? = sin 0 = = = 0

r

1

y

0

tan 0? = tan 0 = = = 0

x

1

p

b. If u = 90? = radians, then the terminal side of the angle is on the positive

2

y@axis. Let us select the point P = (0, 1) with x = 0 and y = 1. This point

is 1 unit from the origin, so r = 1. Figure 4.44 shows values of x, y, and r

p

corresponding to u = 90? or . Now that we know x, y, and r, we can apply

2

the de?nitions of the sine and tangent functions.

y

1

p

sin 90? = sin = = = 1

r

2

1

y

1

p

tan 90? = tan = =

x

2

0

Because division by 0 is unde?ned, tan 90�� is unde?ned.

c. If u = 180? = p radians, then the terminal side of the angle is on the

negative x@axis. Let us select the point P = (-1, 0) with x = -1 and y = 0.

This point is 1 unit from the origin, so r = 1. Figure 4.45 shows values of

x, y, and r corresponding to u = 180? or p. Now that we know x, y, and r,

we can apply the de?nitions of the sine and tangent functions.

y

0

sin 180? = sin p = = = 0

r

1

y

0

tan 180? = tan p = =

= 0

x

-1

3p

d. If u = 270? =

radians, then the terminal side of the angle is on the negative

2

y@axis. Let us select the point P = (0, -1) with x = 0 and y = -1. This

point is 1 unit from the origin, so r = 1. Figure 4.46 shows values of x, y,

3p

and r corresponding to u = 270? or

. Now that we know x, y, and r, we

2

can apply the de?nitions of the sine and tangent functions.

DISCOVERY

y

-1

3p

= =

= -1

Try ?nding tan 90�� and tan 270��

r

2

1

with your calculator. Describe

y

-1

3p

tan 270? = tan

= =

what occurs.

x

2

0

Because division by 0 is unde?ned, tan 270�� is unde?ned.

�� �� ��

sin 270? = sin

540

Chapter 4 Trigonometric Functions

Check Point 2 Evaluate, if possible, the cosine function and the cosecant

function at the following four quadrantal angles:

p

3p

a. u = 0? = 0 b. u = 90? =

c. u = 180? = p d. u = 270? =

.

2

2

???

The Signs of the Trigonometric Functions

Use the signs of the

trigonometric functions.

y

Quadrant II

sine and

cosecant

positive

Quadrant I

All

functions

positive

x

Quadrant III

tangent and

cotangent

positive

Quadrant IV

cosine and

secant

positive

FIGURE 4.47 The signs of the

trigonometric functions

In Example 2, we evaluated trigonometric functions of quadrantal angles. However,

we will now return to the trigonometric functions of nonquadrantal angles. If U is not

a quadrantal angle, the sign of a trigonometric function depends on the quadrant in

which U lies. In all four quadrants, r is positive. However, x and y can be positive or

negative. For example, if u lies in quadrant II, x is negative and y is positive. Thus, the

y

r

only positive ratios in this quadrant are and its reciprocal, . These ratios are the

y

r

function values for the sine and cosecant, respectively. In short, if u lies in quadrant II,

sin u and csc u are positive. The other four trigonometric functions are negative.

Figure 4.47 summarizes the signs of the trigonometric functions. If u lies in

quadrant I, all six functions are positive. If u lies in quadrant II, only sin u and csc u

are positive. If u lies in quadrant III, only tan u and cot u are positive. Finally, if u lies

in quadrant IV, only cos u and sec u are positive. Observe that the positive functions

in each quadrant occur in reciprocal pairs.

GREAT QUESTION!

Is there a way to remember the signs of the trigonometric functions?

Here��s a phrase that may be helpful:

A

Trig

Smart

All trig functions

are positive in

QI.

EXAMPLE 3

Sine and its

reciprocal, cosecant,

are positive in QII.

Tangent and its

reciprocal, cotangent,

are positive in QIII.

Class.

Cosine and its

reciprocal, secant,

are positive in QIV.

Finding the Quadrant in Which an Angle Lies

If tan u 6 0 and cos u 7 0, name the quadrant in which angle u lies.

SOLUTION

When tan u 6 0, u lies in quadrant II or IV. When cos u 7 0, u lies in quadrant

I or IV. When both conditions are met (tan u 6 0 and cos u 7 0), u must lie in

quadrant IV.

�� �� ��

Check Point 3 If sin u 6 0 and cos u 6 0, name the quadrant in which angle

u lies.

EXAMPLE 4

Evaluating Trigonometric Functions

2

3

Given tan u = - and cos u 7 0, ?nd cos u and csc u.

SOLUTION

Because the tangent is negative and the cosine is positive, u lies in quadrant IV.

This will help us to determine whether the negative sign in tan u = - 23 should

be associated with the numerator or the denominator. Keep in mind that in

quadrant IV, x is positive and y is negative. Thus,

In quadrant IV, y is negative.

y

�C2

2

tan u=�C = = .

x

3

3

Section 4.4 Trigonometric Functions of Any Angle

(See Figure 4.48.) Thus, x = 3 and y = -2. Furthermore,

y

5

u

r = 2x2 + y 2 = 232 + (-2)2 = 29 + 4 = 213.

Now that we know x, y, and r, we can ?nd cos u and csc u.

r = ��13

?5

5

?5

541

x

P = (3, ? 2)

x=3

cos u =

3

x

3 # 213

3213

=

=

=

r

13

213

213 213

csc u =

213

r

213

=

= y

-2

2

�� �� ��

Check Point 4 Given tan u = - 13 and cos u 6 0, ?nd sin u and sec u.

y = ?2

2

3

FIGURE 4.48 tan u = - and cos u 7 0

In Example 4, we used the quadrant in which u lies to determine whether a

negative sign should be associated with the numerator or the denominator. Here��s a

situation, similar to Example 4, where negative signs should be associated with both

the numerator and the denominator:

tan u =

3

5

and cos u 6 0.

Because the tangent is positive and the cosine is negative, u lies in quadrant III. In

quadrant III, x is negative and y is negative. Thus,

y

3

�C3

tan u= = =

.

x

5

�C5

???

Find reference angles.

We see that x = ?5

and y = ?3.

Reference Angles

We will often evaluate trigonometric functions of positive angles greater than 90��

and all negative angles by making use of a positive acute angle. This positive acute

angle is called a reference angle.

De?nition of a Reference Angle

Let u be a nonacute angle in standard position that lies in a quadrant. Its reference

angle is the positive acute angle u? formed by the terminal side of u and the x@axis.

Figure 4.49 shows the reference angle for u lying in quadrants II, III, and IV.

Notice that the formula used to ?nd u?, the reference angle, varies according to the

quadrant in which u lies. You may ?nd it easier to ?nd the reference angle for a given

angle by making a ?gure that shows the angle in standard position. The acute angle

formed by the terminal side of this angle and the x@axis is the reference angle.

y

u?

y

u

u

x

x

u?

FIGURE 4.49 Reference angles, u?, for

positive angles, u, in quadrants II, III,

and IV

If 90? ? u ? 180?,

then u ? ? 180? ? u.

y

If 180? ? u ? 270?,

then u ? ? u ? 180?.

u

u?

If 270? ? u ? 360?,

then u ? ? 360? ? u.

x

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