Section 4.4 Trigonometric Functions of Any Angle...

Section 4.4 Trigonometric Functions of Any Angle 537

Objectives � Use the defi nitions of

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 SECTION 4.4

C ycles 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.

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 defi nitions of trigonometric functions of any angle.

y

x

r

u

y

x

P � (x, y)

(a) u lies inquadrant I.

y

x

r

u

y

x

P � (x, y)

(b) u lies inquadrant II.

y

x

r

u

y

x

P � (x, y)

(c) u lies inquadrant III.

(d) u lies inquadrant IV.

y

x

r

u

y

x

P � (x, y)

FIGURE 4.41

Figures 4.41(b) , (c) , and (d) show angles in standard position, but they are not acute. We can extend our defi 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).

M09_BLITXXXX_05_SE_04-hr.indd 537 08/12/12 2:50 PM

538 Chapter 4 Trigonometric Functions

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

the origin, (0, 0), r = 2x2 + y2 cannot be zero. Examine the six trigonometric functions defi ned above. Note that the denominator of the sine and cosine functions is r. Because r � 0, the sine and cosine functions are defi ned for any angle u. This is not true for the other four trigonometric functions. Note that the denominator of

the tangent and secant functions is x: tan u =yx

and sec u =rx

. These functions are

not defi 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 undefi 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

y = 0, and the cotangent and cosecant functions are undefi ned: cot u =xy

and

csc u =ry

. The cotangent and cosecant functions are undefi ned for all quadrantal

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.

SOLUTION 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 + y2 = 2(-3)2 + (-5)2 = 29 + 25 = 234.

Now that we know x, y, and r, we can fi nd the six trigonometric functions of u. Where appropriate, we will rationalize denominators.

sin u =yr=

-5

234= -

5

234# 234

234= -

523434 csc u =

ry=234-5

= - 234

5

cos u =xr=

-3

234= -

3

234# 234

234= -

323434 sec u =

rx=234-3

= - 234

3

tan u =yx=

-5-3

=53 cot u =

xy=

-3-5

=35

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.

Defi nitions of Trigonometric Functions of Any Angle

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 + y2 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 defi ned by the following ratios:

y

rsin u=

x

rcos u=

y

xtan u= , x � 0

x

ycot u= , y � 0.

r

xsec u= , x � 0

r

ycsc u= , y � 0

The ratios in the second columnare the reciprocals of the corresponding ratios in the

first column.

GREAT QUESTION! Is there a way to make it easier for me to remember the defi 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 defi nitions in the box are the right triangle defi nitions of the trigonometric functions. This should make it easier for you to remember the six defi nitions.

u

r

x = −3 y = −5

5

−5

5−5

y

x

P = (−3, −5)

FIGURE 4.42

● ● ●

y

x

r

u

y

x

P � (x, y)

FIGURE 4.41(a) u lies in quadrant I. (repeated)



How do we fi 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 defi 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:

a. u = 0� = 0 b. u = 90� =p

2 c. u = 180� = p d. u = 270� =

3p2

.

SOLUTION 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 rcorresponding to u = 0� or 0 radians. Now that we know x, y, and r, we can apply the defi nitions of the sine and tangent functions.

sin 0� = sin 0 =yr=

01= 0

tan 0� = tan 0 =yx=

01= 0

b. If u = 90� =p

2 radians, then the terminal side of the angle is on the positive

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

corresponding to u = 90� or p

2. Now that we know x, y, and r, we can apply

the defi nitions of the sine and tangent functions.

sin 90� = sin p

2=

yr=

11= 1

tan 90� = tan p

2=

yx=

10

Because division by 0 is undefi ned, tan 90° is undefi 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 defi nitions of the sine and tangent functions.

sin 180� = sin p =yr=

01= 0

tan 180� = tan p =yx=

0-1

= 0

d. If u = 270� =3p2

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

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,

and r corresponding to u = 270� or 3p2

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

can apply the defi nitions of the sine and tangent functions.

sin 270� = sin 3p2

=yr=

-11

= -1

tan 270� = tan 3p2

=yx=

-10

Because division by 0 is undefi ned, tan 270° is undefi ned. ● ● ●

x = 1 y = 0

1−1

y

xP = (1, 0)u = 0�

r = 1

FIGURE 4.43

x = 0 y = 1

1

1

−1

y

x

P = (0, 1)

u = 90�

1

r = 1

FIGURE 4.44

x = −1 y = 0

1−1

y

xP = (−1, 0)

u = 180�

1

1

r = 1

FIGURE 4.45

x = 0 y = −1

1−1

y

x

P = (0, −1)

u = 270�

1r = 1

−1

FIGURE 4.46

DISCOVERY Try fi nding tan 90° and tan 270° with your calculator. Describe what occurs.



Check Point 2 Evaluate, if possible, the cosine function and the cosecant function at the following four quadrantal angles:

a. u = 0� = 0 b. u = 90� =p

2 c. u = 180� = p d. u = 270� =

3p2

.

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

only positive ratios in this quadrant are yr

and its reciprocal, ry

. These ratios are the

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.

� Use the signs of the trigonometric functions.

y

x

Quadrant IIsine andcosecantpositive

Quadrant IAll

functionspositive

Quadrant IIItangent andcotangentpositive

Quadrant IVcosine and

secantpositive

FIGURE 4.47 The signs of the trigonometric functions

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

Here’s a phrase that may be helpful:

Trig

Tangent and itsreciprocal, cotangent,are positive in QIII.

Class.

Cosine and itsreciprocal, secant,

are positive in QIV.

Smart

Sine and itsreciprocal, cosecant,are positive in QII.

A

All trig functionsare positive in

QI.

EXAMPLE 3 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

Given tan u = - 23 and cos u 7 0, fi 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,

tan u= .– = =

2

3

–2

3

y

x

In quadrant IV, y is negative.



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

r = 2x2 + y2 = 232 + (-2)2 = 29 + 4 = 213.

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

cos u =xr=

3

213=

3

213# 213

213=

321313 csc u =

ry=213-2

= - 213

2 ● ● ●

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

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 =35 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,

3

5

–3

–5

y

x= .=tan u=

We see that x = −5and 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.

u

x = 3 y = −2

5

−5

5−5

y

x

P = (3, −2)

r = �13

FIGURE 4.48 tan u = - 23 and cos u 7 0

� Find reference angles.

Defi 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 fi nd u�, the reference angle, varies according to the quadrant in which u lies. You may fi nd it easier to fi nd the reference angle for a given angle by making a fi 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.

If 90� � u � 180�,then u � � 180� � u.

u

u�

y

x

If 180� � u � 270�,then u � � u � 180�.

u

u�

y

x

If 270� � u � 360�,then u � � 360� � u.

u

u�

y

x

FIGURE 4.49 Reference angles, u�, for positive angles, u, in quadrants II, III, and IV



EXAMPLE 5 Finding Reference Angles

Find the reference angle, u�, for each of the following angles:

a. u = 345� b. u =5p6

c. u = -135� d. u = 2.5.

SOLUTION a. A 345° angle in standard position is

shown in Figure 4.50 . Because 345° lies in quadrant IV, the reference angle is

u� = 360� - 345� = 15�.

b. Because 5p6

lies between p

2=

3p6

and

p =6p6

, u =5p6

lies in quadrant II. The

angle is shown in Figure 4.51 . The reference angle is

u� = p -5p6

=6p6

-5p6

=p

6.

c. A -135� angle in standard position is shown in Figure 4.52 . The fi gure indicates that the positive acute angle formed by the terminal side of u and the x@axis is 45°. The reference angle is

u� = 45�.

d. The angle u = 2.5 lies between p

2 � 1.57

and p � 3.14. This means that u = 2.5 is in quadrant II, shown in Figure 4.53 . The reference angle is

u� = p - 2.5 � 0.64.

Check Point 5 Find the reference angle, u�, for each of the following angles:

a. u = 210� b. u =7p4

c. u = -240� d. u = 3.6.

Finding reference angles for angles that are greater than 360� (2p) or less than -360� (-2p) involves using coterminal angles. We have seen that coterminal angles have the same initial and terminal sides. Recall that coterminal angles can be obtained by increasing or decreasing an angle’s measure by an integer multiple of 360° or 2p.

u � 345�

u� � 15�

y

x

FIGURE 4.50

y

xu� � p6

u � 5p6

FIGURE 4.51

x

u � �135�u� � 45�

y

FIGURE 4.52

y

xu� � 0.64

u � 2.5

FIGURE 4.53 ● ● ●

DISCOVERY Solve part (c) by fi rst fi nding a positive coterminal angle for -135� less than 360°. Use the positive coterminal angle to fi nd the reference angle.

Finding Reference Angles for Angles Greater Than 360� (2P) or Less Than �360� (�2P)

1. Find a positive angle a less than 360° or 2p that is coterminal with the given angle.

2. Draw a in standard position. 3. Use the drawing to fi nd the reference angle for the given angle. The positive

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



EXAMPLE 6 Finding Reference Angles

Find the reference angle for each of the following angles:

a. u = 580� b. u =8p3

c. u = - 13p

6.

SOLUTION a. For a 580° angle, subtract 360° to fi nd a positive coterminal angle less than 360°.

580� - 360� = 220�

Figure 4.54 shows a = 220� in standard position. Because 220° lies in quadrant III, the reference angle is

a� = 220� - 180� = 40�.

b. For an 8p3

, or 2 23

p , angle, subtract 2p to fi nd a

positive coterminal angle less than 2p.

8p3

- 2p =8p3

-6p3

=2p3

Figure 4.55 shows a =2p3

in standard

position. Because 2p3

lies in quadrant II, the reference angle is

a� = p -2p3

=3p3

-2p3

=p

3.

c. For a - 13p

6, or -2

16

p , angle, add 4p to fi nd a


- 13p

6+ 4p = -

13p6

+24p

6=

11p6

Figure 4.56 shows a =11p

6 in standard

position. Because 11p

6 lies in quadrant IV, the reference angle is

a� = 2p -11p

6=

12p6

-11p

6=p

6. ● ● ●

Check Point 6 Find the reference angle for each of the following angles:

a. u = 665� b. u =15p

4 c. u = -

11p3

.

Evaluating Trigonometric Functions Using Reference Angles The way that reference angles are defi ned makes them useful in evaluating trigonometric functions.

x

a � 220�

a� � 40�

y

FIGURE 4.54

y

x

a� � p3a � 2p

3

FIGURE 4.55

y

xa� � p6

a � 11p6

FIGURE 4.56

DISCOVERY Solve part (c) using the coterminal angle formed by adding 2p, rather than 4p, to the given angle.

� Use reference angles to evaluate trigonometric functions.

Using Reference Angles to Evaluate Trigonometric Functions

The values of the trigonometric functions of a given angle, u, are the same as the values of the trigonometric functions of the reference angle, u�, except possibly for the sign. A function value of the acute reference angle, u�, is always positive. However, the same function value for u may be positive or negative.



For example, we can use a reference angle, u�, to obtain an exact value for tan 120°. The reference angle for u = 120� is u� = 180� - 120� = 60�. We know the exact value of the tangent function of the reference angle: tan 60� = 23. We also know that the value of a trigonometric function of a given angle, u, is the same as that of its reference angle, u�, except possibly for the sign. Thus, we can conclude that tan 120° equals -23 or 23.

What sign should we attach to 23? A 120° angle lies in quadrant II, where only the sine and cosecant are positive. Thus, the tangent function is negative for a 120° angle. Therefore,

tan 120�=–tan 60�=–23.

Prefix by a negative sign toshow tangent is negative in

quadrant II.

The reference anglefor 120° is 60°.

In the previous section, we used two right triangles to fi nd exact trigonometric values of 30°, 45°, and 60°. Using a procedure similar to fi nding tan 120°, we can now fi nd the exact function values of all angles for which 30°, 45°, or 60° are reference angles.

A Procedure for Using Reference Angles to Evaluate Trigonometric Functions

The value of a trigonometric function of any angle u is found as follows:

1. Find the associated reference angle, u�, and the function value for u�. 2. Use the quadrant in which u lies to prefi x the appropriate sign to the

function value in step 1.

DISCOVERY Draw the two right triangles involving 30°, 45°, and 60°. Indicate the length of each side. Use these lengths to verify the function values for the reference angles in the solution to Example 7.

EXAMPLE 7 Using Reference Angles to Evaluate Trigonometric Functions

Use reference angles to fi nd the exact value of each of the following trigonometric functions:

a. sin 135° b. cos4p3

c. cota- p

3b .

SOLUTION a. We use our two-step procedure to fi nd sin 135°.

Step 1 Find the reference angle, U�, and sin U�. Figure 4.57 shows 135° lies in quadrant II. The reference angle is

u� = 180� - 135� = 45�.

The function value for the reference angle is sin 45� =222

.

Step 2 Use the quadrant in which U lies to prefi x the appropriate sign to the function value in step 1. The angle u = 135� 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,

22

2sin 135�=±sin 45�= .

The sine is positivein quadrant II.

The reference anglefor 135° is 45°.

y

x45�

135�

FIGURE 4.57 Reference angle for 135°



b. We use our two-step procedure to fi nd cos 4p3

.

Step 1 Find the reference angle, U�, and cos U�. Figure 4.58 shows that

u =4p3

lies in quadrant III. The reference angle is

u� =4p3

- p =4p3

-3p3

=p

3.

The function value for the reference angle is

cos p

3=

12

.

Step 2 Use the quadrant in which U lies to prefi x the appropriate sign to

the function value in step 1. The angle u =4p3

lies in quadrant III. Because

only the tangent and cotangent are positive in quadrant III, the cosine is negative in this quadrant. We put a - sign before the function value of the reference angle. Thus,

=–cos4p

3cos

p

3=– .

1

2

The reference angle

The cosine is negativein quadrant III.

for is .4p3p3

c. We use our two-step procedure to fi nd cota- p

3 b .

Step 1 Find the reference angle, U�, and cot U�. Figure 4.59 shows that u = -

p

3 lies in quadrant IV. The reference angle is u� =

p

3. The function

value for the reference angle is cot p

3=233

.

Step 2 Use the quadrant in which U lies to prefi x the appropriate sign to

the function value in step 1. The angle u = - p

3 lies in quadrant IV. Because

only the cosine and secant are positive in quadrant IV, the cotangent is negative in this quadrant. We put a - sign before the function value of the reference angle. Thus,

p

3

p

3cot =–cot– .=–

The cotangent isnegative in quadrant IV.

23

3baThe reference angle

for − is .p3p3 ● ● ●

Check Point 7 Use reference angles to fi nd the exact value of the following trigonometric functions:

a. sin 300° b. tan 5p4

c. seca- p

6b .

In our fi nal example, we use positive coterminal angles less than 2p to fi nd the reference angles.

y

xp

3

4p3

FIGURE 4.58 Reference angle for 4p3

�p

3

y

xp

3

FIGURE 4.59 Reference angle for - p

3



EXAMPLE 8 Using Reference Angles to EvaluateTrigonometric Functions

Use reference angles to fi nd the exact value of each of the following trigonometric functions:

a. tan 14p

3 b. seca-

17p4b .

SOLUTION

a. We use our two-step procedure to fi nd tan 14p

3.

Step 1 Find the reference angle, U�, and tan U�. Because the given angle,

14p

3 or 4

23p, exceeds 2p, subtract 4p to fi nd a positive coterminal angle

less than 2p.

u =14p

3- 4p =

14p3

-12p

3=

2p3

Figure 4.60 shows u =2p3

in standard position. The angle lies in quadrant II.The reference angle is

u� = p -2p3

=3p3

-2p3

=p

3.

The function value for the reference angle is tan p

3= 23.

Step 2 Use the quadrant in which U lies to prefi x the appropriate sign to the

function value in step 1. The coterminal angle u =2p3

lies in quadrant II.

Because the tangent is negative in quadrant II, we put a - sign before the function value of the reference angle. Thus,

=tan 14p

3

2p

3tan

p

3=–tan =– .

The reference angle

The tangent is negativein quadrant II.

for is .2p3p3

23

b. We use our two-step procedure to fi nd seca- 17p

4b .

Step 1 Find the reference angle, U�, and sec U�. Because the given angle,

- 17p

4 or -4

14

p, is less than -2p, add 6p (three multiples of 2p ) to fi nd a


u = - 17p

4+ 6p = -

17p4

+24p

4=

7p4

Figure 4.61 shows u =7p4

in standard position. The angle lies in quadrant IV. The reference angle is

u� = 2p -7p4

=8p4

-7p4

=p

4.

The function value for the reference angle is sec p

4= 22.

y

x

p

3

2p3


y

xp

4

7p4




Step 2 Use the quadrant in which U lies to prefi x the appropriate sign to the

function value in step 1. The coterminal angle u =7p4

lies in quadrant IV. Because

the secant is positive in quadrant IV, we put a + sign before the function value of the reference angle. Thus,

17p

4

7p

4

p

4

The reference anglefor is .7p

4p4

= .22sec =sec sec – =±

The secant ispositive in quadrant IV.

ba

● ● ●

Check Point 8 Use reference angles to fi nd the exact value of each of the following trigonometric functions:

a. cos 17p

6 b. sina-

22p3b .

GREAT QUESTION! I feel overwhelmed by the amount of information required to evaluate trigonometric functions. Can you help me out?

You’re right. Evaluating trigonometric functions like those in Example 8 and Check Point 8 involves using a number of concepts, including fi nding coterminal angles and reference angles, locating special angles, determining the signs of trigonometric functions

in specifi c quadrants, and fi nding the trigonometric functions of special angles a30� =p

6, 45� =

p

4, and 60� =

p

3b . To be successful

in trigonometry, it is often necessary to connect concepts. Here’s an early reference sheet showing some of the concepts you should have at your fi ngertips (or memorized).

�,0 0 �,0 0

�,30 p

6

�,45 p

4

�,60 p

3�,90 p

2�,120 2p3

�,135 3p4

�,150 5p6

�,210 7p6

�,225 5p4�,240 4p

3 �,270 3p2

�,300 5p3

�,315 7p4

�,330 11p6

�,180 p

�,�30 p

6�

�,�45 p

4�

�,�60 p

3��,�90 p

2��,�120 2p

3�

�,�135 3p4�

�,�150 5p6�

�,�210 7p6�

�,�225 5p4�

�,�240 4p3�

�,�270 3p2�

�,�300 5p3�

�,�315 7p4�

�,�330 11p6�

��,�180 p

1

1�2

45�

1

2 �3

60�

30�

Special Right Triangles and Trigonometric Functions of Special Angles

U 30� �P

6 45� �

P

4 60� �

P

3

sin U 12

222

232

cos U 232

222

12

tan U 233

1 23

Trigonometric Functions of Quadrantal Angles

U 0� � 0 90� �P

2 180� � P 270� �

3P2

sin U 0 1 0 -1

cos U 1 0 -1 0

tan U 0 undefi ned 0 undefi ned

y

x

Quadrant IIsine andcosecantpositive

Quadrant IAll

functionspositive

Quadrant IIItangent andcotangentpositive

Quadrant IVcosine and

secantpositive

Using Reference Angles to Evaluate Trigonometric Functions

sin u=cos u=tan u=

sin u�cos u�tan u�

+ or − in determined by thequadrant in which u lies and the sign

of the function in that quadrant.

Degree and Radian Measures of Special and Quadrantal Angles Signs of the Trigonometric Functions



1. Let u be any angle in standard position and let P = (x, y) be any point besides the origin on the terminal side of u. If r = 2x2 + y2 is the distance from (0, 0) to (x, y), the trigonometric functions of u are defi ned as follows:

sin u = csc u = cos u = sec u = tan u = cot u = .

2. Using the defi nitions in Exercise 1, the trigonometric functions that are undefi ned when x = 0 are

and . The trigonometric functionsthat are undefi ned when y = 0 are and . The trigonometric functions that do not depend on the value of r are and .

3. If u lies in quadrant II, and are positive.

4. If u lies in quadrant III, and are positive.

5. If u lies in quadrant IV, and are positive.

6. Let u be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle formed by the side of u and the

-axis. 7. Complete each statement for a positive angle u and its

reference angle u�. a. If 90� 6 u 6 180�, then u� = . b. If 180� 6 u 6 270�, then u� = . c. If 270� 6 u 6 360�, then u� = .

Fill in each blank so that the resulting statement is true.

CONCEPT AND VOCABULARY CHECK

Practice Exercises In Exercises 1–8, a point on the terminal side of angle u is given. Find the exact value of each of the six trigonometric functions of u.

1. (-4, 3) 2. (-12, 5) 3. (2, 3) 4. (3, 7) 5. (3, -3) 6. (5, -5) 7. (-2, -5) 8. (-1, -3)

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefi ned.

9. cos p 10. tan p 11. sec p

12. csc p 13. tan 3p2

14. cos 3p2

15. cot p

2 16. tan

p

2

In Exercises 17–22, let u be an angle in standard position. Name the quadrant in which u lies.

17. sin u 7 0, cos u 7 0 18. sin u 6 0, cos u 7 0 19. sin u 6 0, cos u 6 0 20. tan u 6 0, sin u 6 0 21. tan u 6 0, cos u 6 0 22. cot u 7 0, sec u 6 0

In Exercises 23–34, fi nd the exact value of each of the remaining trigonometric functions of u.

23. cos u = - 35, u in quadrant III 24. sin u = - 12

13, u in quadrant III 25. sin u = 5

13, u in quadrant II 26. cos u = 4

5, u in quadrant IV 27. cos u = 8

17, 270� 6 u 6 360� 28. cos u = 1

3, 270� 6 u 6 360� 29. tan u = - 23, sin u 7 0 30. tan u = - 1

3, sin u 7 0 31. tan u = 4

3, cos u 6 0 32. tan u = 512, cos u 6 0

33. sec u = -3, tan u 7 0 34. csc u = -4, tan u 7 0

In Exercises 35–60, fi nd the reference angle for each angle.

35. 160° 36. 170° 37. 205° 38. 210° 39. 355° 40. 351°

41. 7p4

42. 5p4

43. 5p6

44. 5p7

45. -150� 46. -250�

47. -335� 48. -359� 49. 4.7 50. 5.5 51. 565° 52. 553°

53. 17p

6 54.

11p4

55. 23p

4

56. 17p

3 57. -

11p4

58. - 17p

6

59. - 25p

6 60. -

13p3

In Exercises 61–86, use reference angles to fi nd the exact value of each expression. Do not use a calculator.

61. cos 225° 62. sin 300° 63. tan 210° 64. sec 240° 65. tan 420° 66. tan 405°

67. sin 2p3

68. cos 3p4

69. csc 7p6

70. cot 7p4

71. tan 9p4

72. tan 9p2

73. sin(-240�) 74. sin(-225�) 75. tana-p4b

76. tana-p6b 77. sec 495° 78. sec 510°

79. cot 19p

6 80. cot

13p3

81. cos 23p

4

82. cos 35p

6 83. tana-17p

6b 84. tana-11p

4b

85. sina-17p3b 86. sina-35p

6b

EXERCISE SET 4.4


Mid-Chapter Check Point 549

107. If cos u 7 0 and tan u 6 0, explain how to fi nd the quadrant in which u lies.

108. What is a reference angle? Give an example with your description.

109. Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.

Critical Thinking Exercises Make Sense? In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning.

110. I’m working with a quadrantal angle u for which sin u is undefi ned.

111. This angle u is in a quadrant in which sin u 6 0 and csc u 7 0.

112. I am given that tan u = 35, so I can conclude that y = 3 and

x = 5. 113. When I found the exact value of cos 14p

3 , I used a number of concepts, including coterminal angles, reference angles, fi nding the cosine of a special angle, and knowing the cosine’s sign in various quadrants.

Preview Exercises Exercises 114–116 will help you prepare for the material covered in the next section. In each exercise, complete the table of coordinates. Do not use a calculator.

114. y = 12 cos (4x + p)

x - p4 - p8 0 p8 p4

y

115. y = 4 sin 12x - 2p3 2

x p3 7p12 5p6 13p12 4p3

y

116. y = 3 sin p2 x

x 0 13 1 53 2 73 3 113 4

y

After completing this table of coordinates, plot the nine ordered pairs as points in a rectangular coordinate system. Then connect the points with a smooth curve.

Practice Plus In Exercises 87–92, fi nd the exact value of each expression. Write the answer as a single fraction. Do not use a calculator.

87. sin p

3 cos p - cos

p

3 sin

3p2

88. sin p

4 cos 0 - sin

p

6 cos p

89. sin 11p

4 cos

5p6

+ cos 11p

4 sin

5p6

90. sin 17p

3 cos

5p4

+ cos 17p

3 sin

5p4

91. sin 3p2

tana- 15p4b - cosa-5p

3b

92. sin 3p2

tana- 8p3b + cosa-5p

6b

In Exercises 93–98, let

f(x) = sin x, g(x) = cos x, and h(x) = 2x.

Find the exact value of each expression. Do not use a calculator.

93. f a4p3

+p

6b + f a4p

3b + f ap

6b

94. ga5p6

+p

6b + ga5p

6b + gap

6b

95. (h � g)a17p3b 96. (h � f)a11p

4b

97. the average rate of change of f from x1 =5p4

to x2 =3p2

98. the average rate of change of g from x1 =3p4

to x2 = p

In Exercises 99–104, fi nd two values of u, 0 … u 6 2p, that satisfy each equation.

99. sin u =222

100. cos u =12

101. sin u = - 222

102. cos u = -12

103. tan u = -23 104. tan u = -233

Writing in Mathematics 105. If you are given a point on the terminal side of angle u,

explain how to fi nd sin u. 106. Explain why tan 90° is undefi ned.

Mid-Chapter Check Point CHAPTER 4

WHAT YOU KNOW: We learned to use radians to measure angles: One radian (approximately 57°) is the measure of the central angle that intercepts an arc equal in length to the radius of the circle. Using 180� = p radians,

we converted degrees to radians a multiply by p

180�b and

radians to degrees a multiply by 180�

pb . We defi ned the


six trigonometric functions using coordinates of points along the unit circle, right triangles, and angles in standard position. Evaluating trigonometric functions using reference angles involved connecting a number of concepts, including fi nding coterminal and reference angles, locating special angles, determining the signs of the trigonometric functions in specifi c quadrants, and fi nding the function values at special angles. Use the important Great Question! box on page 547 as a reference sheet to help connect these concepts.

In Exercises 1–2, convert each angle in degrees to radians. Express your answer as a multiple of p.

1. 10° 2. -105�

In Exercises 3–4, convert each angle in radians to degrees.

3. 5p12

4. - 13p20

In Exercises 5–7,

a. Find a positive angle less than 360° or 2p that is coterminal with the given angle.

b. Draw the given angle in standard position.

c. Find the reference angle for the given angle.

5. 11p

3 6. -

19p4

7. 510°

8. Use the point shown on the unit circle to fi nd each of the six trigonometric functions at t.

x2 + y2 = 1

x

y

(1, 0)

t

t

P ,�35 �

45 )(

9. Use the triangle to fi nd each of the six trigonometric functions of u.

65

u

B

CA

10. Use the point on the terminal side of u to fi nd each of the six trigonometric functions of u.

u

y

x

P(3, �2)

In Exercises 11–12, fi nd the exact value of the remaining trigonometric functions of u.

11. tan u = - 34

, cos u 6 0 12. cos u =37

, sin u 6 0

In Exercises 13–14, fi nd the measure of the side of the right triangle whose length is designated by a lowercase letter. Round the answer to the nearest whole number.

13.

60 cm

a

B

CA41�

14.

c250 m

B

CA

72�

15. If cos u =16

and u is acute, fi nd cotap2

- ub .

In Exercises 16–26, fi nd the exact value of each expression. Do not use a calculator.

16. tan 30° 17. cot 120°

18. cos 240° 19. sec 11p

6

20. sin2p

7+ cos2p

7 21. sina-

2p3b

22. csca22p3b 23. cos 495°

24. tana- 17p

6b 25. sin2

p

2- cos p

26. cosa5p6

+ 2pnb + tana5p6

+ npb , n is an integer.

27. A circle has a radius of 40 centimeters. Find the length of the arc intercepted by a central angle of 36°. Express the answer in terms of p. Then round to two decimal places.

28. A merry-go-round makes 8 revolutions per minute. Find the linear speed, in feet per minute, of a horse 10 feet from the center. Express the answer in terms of p. Then round to one decimal place.

29. A plane takes off at an angle of 6°. After traveling for one mile, or 5280 feet, along this fl ight path, fi nd the plane’s height, to the nearest tenth of a foot, above the ground.

30. A tree that is 50 feet tall casts a shadow that is 60 feet long. Find the angle of elevation, to the nearest degree, of the sun.



Date post:	25-Aug-2020
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

Section 4.4 Trigonometric Functions of Any Angle...

Documents