Section 5-4 Trigonometric Functions

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Section 5-4 Trigonometric Functions

Definition of the Trigonometric Functions Calculator Evaluation of Trigonometric Functions

Definition of the Trigonometric Functions—Alternate Form Exact Values for Special Angles and Real Numbers

Summary of Special Angle Values

The six circular functions introduced in Section 5-2 were defined without any ref-erence to the concept of angle. Historically, however, angles and triangles formed the core subject matter of trigonometry. In this section we introduce the six trigonometric functions, whose domain values are the measures of angles. The six trigonometric functions are intimately related to the six circular functions.

Definition of the Trigonometric Functions

We are now ready to define trigonometric functions with angle domains. Since we have already defined the circular functions with real number domains, we can take advantage of these results and define the trigonometric functions with angle domains in terms of the circular functions. To each of the six circular functions we associate a trigonometric function of the same name. If ␪ is an angle, in either radian or degree measure, we assign values to sin ␪, cos ␪, tan ␪, csc ␪, sec ␪, and cot ␪ as given in Definition 1.

TRIGONOMETRIC FUNCTIONS WITH ANGLE DOMAINS

If ␪ is an angle with radian measure x, then the value of each trigono-metric function at ␪ is given by its value at the real number x.

Trigonometric Circular Function Function sin ␪ ⫽ sin x cos ␪ ⫽ cos x tan ␪ ⫽ tan x csc ␪ ⫽ csc x sec ␪ ⫽ sec x cot ␪ ⫽ cot x

If ␪ is an angle in degree measure, convert to radian measure and pro-ceed as above.

[Note: To reduce the number of different symbols in certain figures, the u and v axes we started with will often be labeled as the a and b axes, respectively. Also, an expression such as sin 30° denotes the sine of the angle whose measure is 30°.]

(1, 0) ␪ x rad W(x) (a, b) x units arc length a b

D E F I N I T I O N

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C A U T I O N

The figure in Definition 1 makes use of the important fact that in a unit cir-cle the arc length s opposite an angle of x radians is x units long, and vice versa:

s⫽ r␪ ⫽ 1 ⴢ x ⫽ x

Exact Evaluation for Special Angles Evaluate exactly without a calculator.

(A) (B) (C) cos 180° (D) csc (⫺150°) S o l u t i o n s (A) (B) (C) cos 180° ⫽ cos ␲ ⫽ ⫺1 (D)

Evaluate exactly without a calculator. (A) tan (⫺␲/4 rad) (B) cos (2␲/3 rad)

(C) sin 90° (D) sec (⫺120°)

Calculator Evaluation of Trigonometric Functions

How do we evaluate trigonometric functions for arbitrary angles? Just as a cal-culator can be used to approximate circular functions for arbitrary real numbers, a calculator can be used to approximate trigonometric functions for arbitrary angles.

Most calculators have a choice of three trigonometric modes: degree (deci-mal), radian, or grad.

The measure of a right angle

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T H E O R E M

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Using a calculator with degree and radian modes, we can evaluate trigono-metric functions directly for angles in either degree or radian measure without having to convert degree measure to radian measure first. (Some calculators work only with decimal degrees and others work with either decimal degrees or degree–minute–second forms. Consult your manual.)

We generalize the reciprocal identities (stated first in Theorem 1, Section 5-2) to evaluate cosecant, secant, and cotangent.

RECIPROCAL IDENTITIES

For x any real number or angle in degree or radian measure,

Calculator Evaluation

Evaluate to four significant digits using a calculator.

(A) cos 173.42° (B) sin (3 rad) (C) tan 7.183

(D) cot (⫺102°51⬘) (E) sec (⫺12.59 rad) (E) csc (⫺206.3)

S o l u t i o n s (A) cos 173.42°⫽ ⫺0.9934 Degree mode

(B) sin (3 rad) ⫽ 0.1411 Radian mode

(C) tan 7.183 ⫽ 1.260 Radian mode

(D) cot (⫺102°51⬘)

⫽ 0.2281

(E) sec (⫺12.59 rad) ⫽ 1.000 Radian mode

(F) csc (⫺206.3) ⫽ 1.156 Radian mode

Evaluate to four significant digits using a calculator.

(A) sin 239.12° (B) cos (7 radians) (C) cot 10

(D) tan (⫺212°33⬘) (E) sec (⫺8.09 radians) (F) csc (⫺344.5)

Definition of the Trigonometric Functions

—

Alternate Form

For many applications involving the use of trigonometric functions, including tri-angle applications, it is useful to write Definition 1 in an alternate form—a form

M A T C H E D P R O B L E M

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Degree mode (Some calculators require decimal degrees.)

⫽ cot (⫺102.85°) E X A M P L E

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cot x⫽ 1 tan x tan x⫽ 0 sec x⫽ 1 cos x cos x⫽ 0 csc x⫽ 1 sin x sin x⫽ 0

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that utilizes the coordinates of an arbitrary point (a, b) ⫽ (0, 0) on the terminal side of an angle ␪ (see Fig. 1).

This alternate form of Definition 1 is easily found by inserting a unit circle in Figure 1, drawing perpendiculars from points P and Q to the horizontal axis (Fig. 2), and utilizing the fact that ratios of corresponding sides of similar triangles are proportional.

Letting r⫽ d(O, P) and noting that d(O, Q) ⫽ 1, we have

b and b⬘ always have the same sign. a and a⬘ always have the same sign.

The values of the other four trigonometric functions can be obtained using basic identities. For example,

We now have the very useful alternate form of Definition 1 given below.

TRIGONOMETRIC FUNCTIONS WITH ANGLE DOMAINS

If ␪ is an arbitrary angle in standard position in a rectangular coordinate system and P(a, b) is a point r units from the origin on the terminal side of ␪, then tan ␪ ⫽b a a⫽ 0 cot ␪ ⫽ a b b⫽ 0 cos ␪ ⫽a r sec ␪ ⫽ r a a⫽ 0 sin ␪ ⫽b r csc ␪ ⫽ r b b⫽ 0 b a r ␪ P (a, b) a b r b a ␪ P (a, b) a b P (a, b) r b a ␪ a b tan ␪ ⫽ sin ␪ cos ␪⫽ b/r a/r⫽ b a cos ␪ ⫽ cos x ⫽ a⬘ ⫽a⬘

1 ⫽ a r sin ␪ ⫽ sin x ⫽ b⬘ ⫽b⬘ 1 ⫽ b r P(a, b) is an arbitrary

point on the terminal side of ␪, (a, b) ⫽ (0, 0) r⫽ 兹a2_{⫹ b}2_{⬎ 0;} a b ␪ P(a, b) O FIGURE 1 Angle ␪. ␪ P(a, b) Q(a⬘, b⬘) O x rad (1, 0) x units a b FIGURE 2 Similar triangles.

D E F I N I T I O N

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(Alternate Form)

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Domains: Sets of all possible angles for which the ratios are defined Ranges: Subsets of the set of real numbers

(Domains and ranges will be stated more precisely in the next section.) [Note: The right triangle formed by drawing a perpendicular from P(a, b) to the horizontal axis is called the reference triangle associated with the angle ␪. We will often refer to this triangle.]

Discuss why, for a given angle ␪, the ratios in Definition 1 are indepen-dent of the choice of P(a, b) on the terminal side of ␪ as long as (a, b) ⫽ (0, 0).

The alternate form of Definition 1 should be memorized. As a memory aid, note that when r ⫽ 1, then P(a, b) is on the unit circle, and all function values correspond to the values obtained using Definition 1 for circular functions in Sec-tion 5-2. In fact, using the alternate form of DefiniSec-tion 1 in conjuncSec-tion with the original statement of Definition 1 in this section, we have an alternate way of evaluating circular functions:

CIRCULAR FUNCTIONS AND TRIGONOMETRIC FUNCTIONS For x any real number,

sin x ⫽ sin (x radians) cos x ⫽ cos (x radians)

sec x⫽ sec (x radians) csc x⫽ csc (x radians) (1)

tan x ⫽ tan (x radians) cot x⫽ cot (x radians)

Thus, we are now free to evaluate circular functions in terms of trigonomet-ric functions, using reference triangles where appropriate, or in terms of circular points and the wrapping function discussed earlier. Each approach has certain advantages in particular situations, and you should become familiar with the uses of both approaches.

It is because of equations (1) that we are able to evaluate circular functions using a calculator set in radian mode (see Section 5-2). Generally, unless a cer-tain emphasis is desired, we will not use “rad” after a real number. That is, we will interpret expressions such as “sin 5.73” as the “circular function value sin 5.73” or the “trigonometric function value sin (5.73 rad)” by the context in which the expression occurs or the form we wish to emphasize. We will remain flexible and often switch back and forth between circular function emphasis and trigono-metric function emphasis, depending on which approach provides the most enlightenment for a given situation.

D E F I N I T I O N

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Evaluating Trigonometric Functions

Find the value of each of the six trigonometric functions for the illustrated angle ␪ with terminal side that contains P(⫺3, ⫺4) (see Fig. 3).

S o l u t i o n (a, b) ⫽ (⫺3, ⫺4)

Find the value of each of the six trigonometric functions if the terminal side of ␪ contains the point (⫺6, ⫺8). [Note: This point lies on the terminal side of the angle in Example 3; hence, the final results should be the same as those obtained in Example 3.]

Evaluating Trigonometric Functions

Find the value of each of the other five trigonometric functions for an angle ␪ (without finding ␪) given that ␪ is a IV quadrant angle and sin ␪ ⫽ ⫺ .

S o l u t i o n The information given is sufficient for us to locate a reference triangle in quad-rant IV for ␪, even though we don’t know what ␪ is. We sketch a reference tri-angle, label what we know (Fig. 4), and then complete the problem as indicated.

Since sin ␪ ⫽ b/r ⫽ ⫺ , we can let b ⫽ ⫺4 and

r⫽ 5 (r is never negative). If we can find a,

then we can determine the values of the other five functions. 4 5 a b ⫺5 5 ⫺5 5 a P (a, ⫺4) ⫺4 5 Terminal side of ␪ FIGURE 4 4 5 E X A M P L E

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tan ␪ ⫽b a⫽ ⫺4 ⫺3⫽ 4 3 cot ␪ ⫽ a b⫽ ⫺3 ⫺4⫽ 3 4 cos ␪ ⫽a r ⫽ ⫺3 5 ⫽ ⫺ 3 5 sec ␪ ⫽ r a⫽ 5 ⫺3⫽ ⫺ 5 3 sin ␪ ⫽b r ⫽ ⫺4 5 ⫽ ⫺ 4 5 csc ␪ ⫽ r b⫽ 5 ⫺4⫽ ⫺ 5 4 r⫽ 兹a2⫹ b2⫽ 兹(⫺3)2⫹ (⫺4)2⫽ 兹25 ⫽ 5 a b ⫺5 5 ⫺5 5 r ␪ P (⫺3, ⫺4) FIGURE 3 E X A M P L E

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Use the Pythagorean theorem to find a:

a cannot be negative because ␪ is a IV quadrant

angle.

Using (a, b) ⫽ (3, ⫺4) and r ⫽ 5, we have

Find the value of each of the other five trigonometric functions for an angle ␪ (without finding ␪) given that ␪ is a II quadrant angle and tan ␪ ⫽ ⫺ .

Exact Values for Special Angles and Real Numbers

Assuming a trigonometric function is defined, it can be evaluated exactly with-out the use of a calculator (which is different from finding approximate values using a calculator) for any integer multiple of 30°, 45°, 60°, 90°, ␲/6, ␲/4, ␲/3, or ␲/2. With a little practice you will be able to determine these values mentally. Working with exact values has advantages over working with approximate values in many situations.

The easiest angles to deal with are quadrantal angles since these angles are integer multiples of 90° or ␲/2. It is easy to find the coordinates of a point on a coordinate axis. Since any nonorigin point will do, we shall, for convenience, choose points 1 unit from the origin, as shown in Figure 5.

In each case, , a positive number.

Trig Functions of Quadrantal Angles Find

(A) sin 90° (B) cos ␲ (C) tan (⫺2␲) (D) cot (⫺180°) E X A M P L E

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r⫽ 兹a2_{⫹ b}2_{⫽ 1} (0, 1) (0, ⫺1) (⫺1, 0) (1, 0) a b FIGURE 5 Quadrantal angles. 3 4 M A T C H E D P R O B L E M

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tan ␪ ⫽b a⫽ ⫺4 3 ⫽ ⫺ 4 3 cot ␪ ⫽ a b⫽ 3 ⫺4⫽ ⫺ 3 4 cos ␪ ⫽a r ⫽ 3 5 sec ␪ ⫽ r a⫽ 5 3 csc ␪ ⫽ r b⫽ 5 ⫺4⫽ ⫺ 5 4 ⫽ 3 a⫽ ⫾3 a2_{⫽ 9} a2_{⫹ (⫺4)}2_{⫽ 5}2

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S o l u t i o n s For each, visualize the location of the terminal side of the angle relative to Figure 5. With a little practice, you should be able to do most of the following mentally. (A) (B) (C) (D) Not defined Find

(A) sin (3␲/2) (B) sec (⫺␲) (C) tan 90° (D) cot (⫺270°)

Notice in Example 5, part D, cot (⫺180°) is not defined. Discuss other angles in degree measure for which the cotangent is not defined. For what angles in degree measure is the cosecant function not defined?

Because the concept of reference triangle introduced in Definition 1 (alternate form) plays an important role in much of the material that follows, we restate its definition here and define the related concept of reference angle.

REFERENCE TRIANGLE AND REFERENCE ANGLE

1. To form a reference triangle for ␪, draw a perpendicular from a point P(a, b) on the terminal side of ␪ to the horizontal axis.

2. The reference angle ␣ is the acute angle (always taken positive) between the terminal side of ␪ and the horizontal axis.

(a, b) ⫽ (0, 0) ␣ is always positive a b a ␣ b ␪ P (a, b) M A T C H E D P R O B L E M

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a b (a, b)⫽ (⫺1, 0), r ⫽ 1 ⫽⫺1 0 ⫽ a b cot (⫺180°) a b (a, b)⫽ (1, 0), r ⫽ 1 ⫽0 1⫽ 0 ⫽ b a tan (⫺2␲) a b (a, b)⫽ (⫺1, 0), r ⫽ 1 ⫽⫺1 1 ⫽ ⫺1 ⫽ a r cos ␲ a b (a, b)⫽ (0, 1), r ⫽ 1 ⫽1 1⫽ 1 ⫽ b r sin 90°

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Figure 6 shows several reference triangles and reference angles corresponding to particular angles.

(a) (b)

(c) (d)

(e) (f)

If a reference triangle of a given angle is a 30°–60° right triangle or a 45° right triangle, then we can find exact coordinates, other than (0, 0), on the ter-minal side of the given angle. To this end, we first note that a 30°–60° triangle forms half of an equilateral triangle, as indicated in Figure 7. Because all sides are equal in an equilateral triangle, we can apply the Pythagorean theorem to obtain a useful relationship among the three sides of the original triangle:

␣ ⫽7␲ 6 ⫺ ␲ ⫽ ␲ 6 ␣ ⫽ 420° ⫺ 360° ⫽ 60° ␣ ␪ ⫽ ⫺7␲/6 ⫺␲ ␣ ␪ ⫽ 420⬚ 180⬚ 360⬚ ␣ ⫽ ␪ ⫽␲ 6 ␣ ⫽5␲ 4 ⫺ ␲ ⫽ ␲ 4 ␪ ⫽ ␲/6 ⫽ ␣ ␲/2 ␪ ␣ ␲ ␪ ⫽ 5␲/4 ␣ ⫽

ⱍ

␪

ⱍ

⫽ 45° ␣ ⫽ 180° ⫺ 120° ⫽ 60° ␣ ␪ ⫽ ⫺45⬚ Reference triangle Reference angle ⫺90⬚ ␣ ␪ ⫽ 120⬚ 180⬚ Reference triangle Reference angle FIGURE 6

Reference triangles and refer-ence angles.

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Similarly, using the Pythagorean theorem on a 45° right triangle, we obtain the result shown in Figure 8.

Figure 9 illustrates the results shown in Figures 7 and 8 for the case a ⫽ 1. This case is the easiest to remember. All other cases can be obtained from this special case by multiplying or dividing the length of each side of a triangle in Figure 9 by the same nonzero quantity. For example, if we wanted the hypotenuse of a special 45° right triangle to be 1, we would simply divide each side of the 45° triangle in Figure 9 by .

30°–60° AND 45° SPECIAL TRIANGLES

If an angle or a real number has a 30°–60° or a 45° reference triangle, then we can use Figure 9 to find exact coordinates of a nonorigin point on the

termi-兹2 45⬚ (␲/4) 45⬚ (␲/4) ₁ 1 兹3 30⬚ (␲/6) 60⬚ (␲/3) 2 1 FIGURE 9 兹2 45⬚ (␲/4) 45⬚ (␲/4) a兹2 a a 45⬚ 45⬚ a a c FIGURE 8 45° right triangle. 30⬚ (␲/6) 60⬚ (␲/3) a 2a a兹3 30⬚ 30⬚ 60⬚ 60⬚ a a b c c c FIGURE 7 30°–60° right triangle. ⫽ 兹3a2_{⫽ a兹3} ⫽ 兹(2a)2_{⫺ a}2 b⫽ 兹c2_{⫺ a}2 c⫽ 2a ⫽ a兹2 ⫽ 兹2a2 c⫽ 兹a2⫹ a2

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nal side of the angle. Using the definition of the trigonometric functions, Defini-tion 1 alternate form, we will then be able to find the exact value of any of the six functions for the indicated angle or real number.

Exact Evaluation

Evaluate exactly using appropriate reference triangles.

(A) cos 60°, sin (␲/3), tan (␲/3) (B) sin 45°, cot (␲/4), sec (␲/4)

S o l u t i o n s (A) Use the special 30°–60° triangle with sides 1, 2, and as the reference triangle, and use 60° or ␲/3 as the reference angle (Fig. 10). Use the sides of the reference triangle to determine P(a, b) and r; then use the appropri-ate definitions.

(B) Use the special 45° triangle with sides 1, 1, and as the reference trian-gle, and use 45° or ␲/4 as the reference angle (Fig. 11). Use the sides of the reference triangle to determine P(a, b) and r; then use the appropriate definitions.

(A) cos 45°, tan (␲/4), csc (␲/4) (B) sin 30°, cos (␲/6), cot (␲/6)

Before proceeding, it is useful to observe from a geometric point of view mul-tiples of ␲/3 (60°), ␲/6 (30°), and ␲/4 (45°). These are illustrated in Figure 12.

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sec ␲ 4 ⫽ r a⫽ 兹 2 1 ⫽ 兹2 cot ␲ 4 ⫽ a b⫽ 1 1⫽ 1 sin 45°⫽b r ⫽ 1 兹2 or 兹2 2 a b 45⬚ (␲/4) 兹2 (a, b) ⫽ (1, 1) 1 1 r ⫽ 兹2 FIGURE 11 兹2 tan ␲ 3 ⫽ b a⫽ 兹 3 1 ⫽ 兹3 sin ␲ 3 ⫽ b r⫽ 兹 3 2 cos 60°⫽a r⫽ 1 2 a b 60⬚ (␲/3) 兹3 (a, b) ⫽ (1, 兹3) 1 2 r ⫽ 2 FIGURE 10 兹3 E X A M P L E

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FIGURE 12

Multiples of special angles.

Exact Evaluation

(A) cos (7␲/4) (B) sin (2␲/3) (C) tan 210° (D) sec (⫺240°)

S o l u t i o n s Each angle (or real number) has a 30°–60° or 45° reference triangle. Locate it, determine (a, b) and r, as in Example 6, and then evaluate.

(A) (B) (C) (D) a b ⫺1 2 (a, b) ⫽ (⫺1, 兹3) 兹3 60⬚ ⫺240⬚ r ⫽ 2 a b ⫺1 ₂ (a, b) ⫽ (⫺兹3, ⫺1) ⫺兹3 210⬚ 30⬚ r ⫽ 2 sec (⫺240°) ⫽ 2 ⫺1⫽ ⫺2 tan 210°⫽ ⫺1 ⫺兹3⫽ 1 兹3 or 兹3 3 a b 2␲ 3 ␲ 3 (a, b) ⫽ (⫺1, 兹3) 兹3 2 ⫺1 r ⫽ 2 a b 7␲ 4 ␲ 4 1 ⫺1 (a, b) ⫽ (1, ⫺1) 兹2 r ⫽ 兹2 sin 2␲ 3 ⫽ 兹3 2 cos 7␲ 4 ⫽ 1 兹2 or 兹2 2 E X A M P L E

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0 3␲ 4 5␲ 4 7␲ 4 8␲ 4 ⫽ 2␲ (c) Multiples of ␲/4 (45⬚) 2␲ 4 ␲ 2 ⫽ ␲ 4 6␲ 4 3␲ 2 ⫽ 4␲ 4 ⫽ ␲ 0 7␲ 6 11␲₆ 5␲ 6 ␲ 6 (b) Multiples of ␲/6 (30⬚) 8␲ 6 4␲ 3 ⫽ 4␲ 6 2␲ 3 ⫽ 2␲₆ ⫽␲₃ 3␲ 6 ␲ 2 ⫽ 10␲ 6 5␲ 3 ⫽ 12␲ 6 ⫽ 2␲ 6␲ 6 ⫽ ␲ 9␲ 6 3␲ 2 ⫽ 0 2␲ 3 4␲ 3 5␲ 3 ␲ 3 3␲ 3 ⫽ ␲ (a) Multiples of ␲/3 (60⬚) 6␲ 3 ⫽ 2␲

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Evaluate exactly using appropriate triangles.

(A) tan (⫺␲/4) (B) sin 210° (C) cos (2␲/3) (D) csc (⫺240°)

Now we reverse the problem; that is, we let the exact value of one of the six trigonometric functions be given and assume this value corresponds to one of the special reference triangles. Can we find a smallest positive ␪ for which the trigonometric function has that value? Example 8 shows how.

Finding ␪

Find the smallest positive ␪ in degree and radian measure for which each is true.

(A) (B)

S o l u t i o n s (A)

We can let (a, b) ⫽ ( , 1) or (⫺ , ⫺1). The smallest positive ␪ for which this is true is a quadrant I angle with reference triangle as drawn in Figure 13.

(B) Because r ⬎ 0

In quadrants II and III, a is negative. The smallest positive ␪ is associated with a 45° reference triangle in quadrant II, as drawn in Figure 14.

␪ ⫽ 135° or 3␲ 4 a b ⫺1 兹2 45⬚ 1 ␪ FIGURE 14 sec ␪ ⫽ r a⫽ 兹 2 ⫺1 ␪ ⫽ 30° or ␲ 6 兹3 1 30⬚ (兹3, 1) a b FIGURE 13 兹3 兹3 tan ␪ ⫽ b a⫽ 1 兹3 sec ␪ ⫽ ⫺兹2 tan ␪ ⫽ 1/兹3 E X A M P L E

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E x p l o r e / D i s c u s s

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Find the smallest positive ␪ in degree and radian measure for which each is true.

(A) (B)

Remark After quite a bit of practice, the reference triangle figures in Examples 7 and 8 can be visualized mentally; however, when in doubt, draw a figure.

Summary of Special Angle Values

Table 1 includes a summary of the exact values of the sine, cosine, and tangent for the special angle values from 0° to 90°. Some people like to memorize these values, while others prefer to memorize the triangles in Figure 9. Do whichever is easier for you.

␪ sin ␪ cos ␪ tan ␪

0° 0 1 0

30°

45° 1

60°

90° 1 0 Not defined

These special angle values are easily remembered for sine and cosine if you note the unexpected pattern after completing Table 2 in Explore/Discuss 3.

Fill in the cosine column in Table 2 with a pattern of values that is simi-lar to those in the sine column. Discuss how the two columns of values are related. ␪ sin ␪ cos ␪ 0° 30° 45° 60° 90°

Cosecant, secant, and cotangent can be found for these special angles by using the values in Tables 1 or 2 and the reciprocal identities from Theorem 1.

兹4/2 ⫽ 1 兹3/2 兹2/2 兹1/2 ⫽1 2 兹0/2 ⫽ 0

T A B L E 2 Special Angle Values——Memory Aid 兹3 1 2 兹3/2 1/兹2 or 兹2/2 1/兹2 or 兹2/2 1/兹3 or 兹3/3 兹3/2 1 2

T A B L E 1 Special Angle Values

cos ␪ ⫽ ⫺1/兹2 sin ␪ ⫽ 兹3/2

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A n s w e r s t o M a t c h e d P r o b l e m s

1. (A) ⫺1 (B) (C) 1 (D) ⫺2

2. (A) ⫺0.8582 (B) 0.7539 (C) 1.542 (D) ⫺0.6383 (E) ⫺4.277 (F) 1.137 3.

4.

5. (A) ⫺1 (B) ⫺1 (C) Not defined (D) 0

6. (A) (B) 7. (A) ⫺1 (B) (C) (D) 8. (A) 60° or ␲/3 (B) 135° or 3␲/4 2/兹3 ⫺1 2 ⫺1 2

sin 30°⫽1₂, cos (␲/6) ⫽ 兹3/2, cot (␲/6) ⫽ 兹3 cos 45°⫽ 1/兹2, tan (␲/4) ⫽ 1, csc (␲/4) ⫽ 兹2

sin ␪ ⫽3₅, cos ␪ ⫽ ⫺4₅, csc ␪ ⫽5₃, sec ␪ ⫽ ⫺5₄, cot ␪ ⫽ ⫺4₃

sin ␪ ⫽ ⫺4₅, cos ␪ ⫽ ⫺3₅, tan ␪ ⫽4₃, csc ␪ ⫽ ⫺5₄, sec ␪ ⫽ ⫺5₃, cot ␪ ⫽3₄ ⫺1

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E X E R C I S E 5 - 4

A

Find the value of each of the six trigonometric functions for an angle ␪ that has a terminal side containing the point indicated in Problems 1–4.

1. (6, 8) 2. (⫺3, 4)

3. 4.

Evaluate Problems 5–14 to four significant digits using a calculator. Make sure your calculator is in the correct mode (degree or radian) for each problem.

5. sin 25° 6. tan 89° 7. cot 12 8. csc 13 9. sin 2.137 10. tan 4.327 11. cot (⫺431.41°) 12. sec (⫺247.39°) 13. sin 113°27⬘13⬙ 14. cos 235°12⬘47⬙

In Problems 15–26, evaluate exactly, using reference triangles where appropriate, without using a calculator.

15. sin 0° 16. cos 0° 17. tan 60° 18. cos 30° 19. sin 45° 20. csc 60° 21. sec 45° 22. cot 45° 23. cot 0° 24. cot 90° 25. tan 90° 26. sec 0°

Find the reference angle ␣ for each angle ␪ in Problems 27–32. 27. ␪ ⫽ 300° 28. ␪ ⫽ 135° 29. 30. 31. 32. ␪ ⫽ ⫺5␲ 4 ␪ ⫽ ⫺5␲ 3 ␪ ⫽␲ 4 ␪ ⫽7␲ 6 (兹3, 1) (⫺1, 兹3)

B

In Problems 33–48, evaluate exactly, using reference angles where appropriate, without using a calculator.

33. cos 120° 34. sin 150° 35. cos (3␲/2) 36. sin (␲/2) 37. cot (⫺60°) 38. sec (⫺30°) 39. cos (⫺␲/6) 40. cot (⫺␲/4) 41. sin (3␲/4) 42. cos (2␲/3) 43. csc 150° 44. cot 225° 45. tan (⫺4␲/3) 46. sec (11␲/6) 47. cos 510° 48. tan 690°

For which values of ␪ 0° ⱕ ␪ ⬍ 360°, is each of Problems 49–54 not defined? Explain why.

49. cos ␪ 50. sec ␪ 51. tan ␪

52. cot ␪ 53. csc ␪ 54. sin ␪

In Problems 55–60, find the smallest positive ␪ in degree and radian measure for which

55. 56. 57.

58. 59. 60.

Find the value of each of the other five trigonometric functions for an angle ␪, without finding ␪, given the information indicated in Problems 61–64. Sketching a reference triangle should be helpful.

61. 62.

63. cos ␪ ⫽ ⫺兹5/3 and cot ␪ ⬎ 0 tan ␪ ⫽ ⫺4₃ and sin ␪ ⬍ 0 sin ␪ ⫽3₅ and cos ␪ ⬍ 0

sec ␪ ⫽ ⫺兹2 csc ␪ ⫽⫺2 兹3 tan ␪ ⫽ ⫺兹3 sin ␪ ⫽⫺1 2 sin ␪ ⫽⫺兹3 2 cos ␪ ⫽⫺1 2

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64.

65. Which trigonometric functions are not defined when the terminal side of an angle lies along the vertical axis. Why?

66. Which trigonometric functions are not defined when the ter-minal side of an angle lies along the horizontal axis? Why? 67. Find exactly, all ␪, 0° ⱕ ␪ ⬍ 360°, for which cos ␪ ⫽

⫺ .

68. Find exactly, all ␪, 0° ⱕ ␪ ⬍ 360°, for which cot ␪ ⫽

⫺ .

69. Find exactly, all ␪, 0 ⱕ ␪ ⬍ 2␲, for which tan ␪ ⫽ 1. 70. Find exactly, all ␪, 0 ⱕ ␪ ⬍ 2␲, for which sec ␪ ⫽ ⫺ .

C

71. If the coordinates of A are (4, 0) and arc length s is 7 units, find

(A) The exact radian measure of ␪

(B) The coordinates of P to three decimal places

72. If the coordinates of A are (2, 0) and arc length s is 8 units, find

(A) The exact radian measure of ␪

(B) The coordinates of P to three decimal places 73. In a rectangular coordinate system, a circle with center at

the origin passes through the point . What is the length of the arc on the circle in quadrant I between the positive horizontal axis and the point ? 74. In a rectangular coordinate system, a circle with center at

the origin passes through the point . What is the length of the arc on the circle in quadrant I between the positive horizontal axis and the point ?

A P P L I C AT I O N S

75. Solar Energy. The intensity of light I on a solar cell changes with the angle of the sun and is given by the for-mula I⫽ k cos ␪, where k is a constant (see the figure).

(2, 2兹3) (2, 2兹3) (6兹3, 6) (6兹3, 6) s A ␪ P (a, b) 兹2 1/兹3 兹3/2

cos ␪ ⫽ ⫺兹5/3 and tan ␪ ⬎ 0 Find light intensity I in terms of k for ␪ ⫽ 0°, ␪ ⫽ 30°, and ␪ ⫽ 60°.

76. Solar Energy. Refer to Problem 75.

Find light intensity I in terms of k for ␪ ⫽ 20°, ␪ ⫽ 50°, and ␪ ⫽ 90°.

77. Physics——Engineering. The figure illustrates a piston connected to a wheel that turns 3 revolutions per second; hence, the angle ␪ is being generated at 3(2␲) ⫽ 6␲ radi-ans per second, or ␪ ⫽ 6␲t, where t is time in seconds. If

P is at (1, 0) when t⫽ 0, show that

for tⱖ 0.

78. Physics——Engineering. In Problem 77, find the position of the piston y when t⫽ 0.2 second (to three significant digits). x y 3 revolutions per second a b ␪ ␪ ⫽ 6␲t P (a, b) 4 inches y (1, 0) ⫽ sin 6␲t ⫹ 兹16 ⫺ (cos 6␲t)2 y⫽ b ⫹ 兹42⫺ a2 ␪ Sun Solar cell

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★79. Geometry. The area of a rectangular n-sided polygon cir-cumscribed about a circle of radius 1 is given by

(A) Find A for n⫽ 8, n ⫽ 100, n ⫽ 1,000, and n ⫽ 10,000. Compute each to five decimal places. (B) What number does A seem to approach as n→⬁?

(What is the area of a circle with radius 1?)

★80. Geometry. The area of a regular n-sided polygon in-scribed in a circle of radius 1 is given by

(A) Find A for n⫽ 8, n ⫽ 100, n ⫽ 1,000, and

n⫽ 10,000. Compute each to five decimal places.

(B) What number does A seem to approach as n→⬁? (What is the area of a circle with radius 1?) 81. Angle of Inclination. Recall (Section 2-1) the slope of a

nonvertical line passing through points P1(x1, y1) and

A⫽n 2 sin 360° n A⫽ n tan 180° n r ⫽ 1 n ⫽ 8

P2(x2, y2) is given by slope ⫽ m ⫽ (y2⫺ y1)/(x2⫺ x1). The angle ␪ that the line L makes with the x axis, 0° ⱕ ␪ ⬍ 180°, is called the angle of inclination of the line L (see figure). Thus,

Slope ⫽ m ⫽ tan ␪, 0° ⱕ ␪ ⬍ 180°

(A) Compute the slopes to two decimal places of the lines with angles of inclination 88.7° and 162.3°.

(B) Find the equation of a line passing through (⫺4, 5) with an angle of inclination 137°. Write the answer in the form y⫽ mx ⫹ b, with m and b to two decimal places.

82. Angle of Inclination Refer to Problem 81.

(A) Compute the slopes to two decimal places of the lines with angles of inclination 5.34° and 92.4°.

(B) Find the equation of a line passing through (6, ⫺4) with an angle of inclination 106°. Write the answer in the form y⫽ mx ⫹ b, with m and b to two decimal places. ␪ ␪ L L x y

Section 5-5 Solving Right Triangles*

In the previous sections we have applied trigonometric and circular functions in the solutions of a variety of significant problems. In this section we are interested in the particular class of problems involving right triangles. A right triangle is a triangle with one 90° angle. Referring to Figure 1, our objective is to find all unknown parts of a right triangle, given the measure of two sides or the measure of one acute angle and a side. This is called solving a right triangle. Trigono-metric functions play a central role in this process.

To start, we locate a right triangle in the first quadrant of a rectangular coor-dinate system and observe, from the definition of the trigonometric functions, six trigonometric ratios involving the sides of the triangle. [Note that the right trian-gle is the reference triantrian-gle for the antrian-gle ␪.]

␤ c b a ␣ FIGURE 1

*This section provides a significant application of trigonometric functions to real-world problems. How-ever, it may be postponed or omitted without loss of continuity, if desired. Some may want to cover the section just before Sections 7-1 and 7-2.