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363 5-4 Trigonometric Functions 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 reference 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. DEFINITION 1 TRIGONOMETRIC FUNCTIONS WITH ANGLE DOMAINS If is an angle with radian measure x, then the value of each trigonometric function at is given by its value at the real number x. Trigonometric Function Circular Function sin ⫽ sin x cos ⫽ cos x tan ⫽ tan x csc ⫽ csc x sec ⫽ sec x cot ⫽ cot x b (a, b) W(x) x units arc length x rad (1, 0) a If is an angle in degree measure, convert to radian measure and proceed 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°.] 364 5 TRIGONOMETRIC FUNCTIONS The figure in Definition 1 makes use of the important fact that in a unit circle the arc length s opposite an angle of x radians is x units long, and vice versa: s ⫽ r ⫽ 1 ⴢ x ⫽ x EXAMPLE Exact Evaluation for Special Angles 1 Evaluate exactly without a calculator. 3 rad rad (A) sin (B) tan 6 4 (C) cos 180° (D) csc (⫺150°) 冢 Solutions 冣 冢 冣 冢6 rad冣 ⫽ sin 6 ⫽ 21 3 3 tan 冢 rad 冣 ⫽ tan ⫽ ⫺1 4 4 (A) sin (B) (C) cos 180° ⫽ cos ( rad) ⫽ cos ⫽ ⫺1 冢 (D) csc (⫺150°) ⫽ csc ⫺ MATCHED PROBLEM 1 5 rad 6 冣 冢 56 冣 ⫽ ⫺2 ⫽ csc ⫺ 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 calculator 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 (decimal), radian, or grad. The measure of a right angle ⫽ 90° ⫽ radians ⫽ 100 grads 2 The grad unit is used in certain engineering applications and will not be used in this book. We repeat a caution stated earlier: CAUTION Read the instruction book accompanying your calculator to determine how to put your calculator in degree or radian mode. Forgetting to set the correct mode before starting calculations involving trigonometric functions is a frequent cause of error when using a calculator. 5-4 Trigonometric Functions 365 Using a calculator with degree and radian modes, we can evaluate trigonometric 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. THEOREM 1 EXAMPLE 2 Solutions MATCHED PROBLEM 2 RECIPROCAL IDENTITIES For x any real number or angle in degree or radian measure, csc x ⫽ 1 sin x sin x ⫽ 0 sec x ⫽ 1 cos x cos x ⫽ 0 cot x ⫽ 1 tan x tan x ⫽ 0 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) (A) cos 173.42° ⫽ ⫺0.9934 (B) sin (3 rad) ⫽ 0.1411 (C) tan 7.183 ⫽ 1.260 (D) cot (⫺102°51⬘) ⫽ cot (⫺102.85°) Degree mode Radian mode Radian mode Degree mode (Some calculators require decimal degrees.) ⫽ 0.2281 (E) sec (⫺12.59 rad) ⫽ 1.000 (F) csc (⫺206.3) ⫽ 1.156 Radian mode 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 triangle applications, it is useful to write Definition 1 in an alternate form—a form 366 5 TRIGONOMETRIC FUNCTIONS 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 FIGURE 1 Angle . b P(a, b) a O sin ⫽ sin x ⫽ b⬘ ⫽ b⬘ b ⫽ 1 r FIGURE 2 Similar triangles. cos ⫽ cos x ⫽ a⬘ ⫽ b b and b⬘ always have the same sign. a⬘ a ⫽ 1 r a and a⬘ always have the same sign. Q(a⬘, b⬘) P(a, b) The values of the other four trigonometric functions can be obtained using basic identities. For example, x units x rad O (1, 0) a tan ⫽ sin b/r b ⫽ ⫽ cos a/r a We now have the very useful alternate form of Definition 1 given below. DEFINITION 1 (Alternate Form) 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 b b b a P (a, b) a r b b a a r a a b r P (a, b) P (a, b) sin ⫽ b r csc ⫽ r b b⫽0 cos ⫽ a r sec ⫽ r a a⫽0 tan ⫽ b a cot ⫽ a b b⫽0 a⫽0 r ⫽ 兹a2 ⫹ b2 ⬎ 0; P(a, b) is an arbitrary point on the terminal side of , (a, b) ⫽ (0, 0) 367 5-4 Trigonometric Functions DEFINITION 1 continued Explore/Discuss 1 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 independent 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 Section 5-2. In fact, using the alternate form of Definition 1 in conjunction 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) tan x ⫽ tan (x radians) cot x ⫽ cot (x radians) (1) Thus, we are now free to evaluate circular functions in terms of trigonometric 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 certain 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 trigonometric function emphasis, depending on which approach provides the most enlightenment for a given situation. 368 5 TRIGONOMETRIC FUNCTIONS EXAMPLE Evaluating Trigonometric Functions 3 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). b FIGURE 3 5 a ⫺5 5 r P (⫺3, ⫺4) ⫺5 (a, b) ⫽ (⫺3, ⫺4) Solution r ⫽ 兹a2 ⫹ b2 ⫽ 兹(⫺3)2 ⫹ (⫺4)2 ⫽ 兹25 ⫽ 5 MATCHED PROBLEM 3 b ⫺4 4 ⫽ ⫽⫺ r 5 5 csc ⫽ r 5 5 ⫽ ⫽⫺ b ⫺4 4 cos ⫽ a ⫺3 3 ⫽ ⫽⫺ r 5 5 sec ⫽ r 5 5 ⫽ ⫽⫺ a ⫺3 3 tan ⫽ b ⫺4 4 ⫽ ⫽ a ⫺3 3 cot ⫽ a ⫺3 3 ⫽ ⫽ b ⫺4 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.] EXAMPLE Evaluating Trigonometric Functions 4 Solution sin ⫽ 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 ⫽ ⫺45. The information given is sufficient for us to locate a reference triangle in quadrant IV for , even though we don’t know what is. We sketch a reference triangle, label what we know (Fig. 4), and then complete the problem as indicated. FIGURE 4 b 5 a ⫺5 5 5 a ⫺4 P (a, ⫺4) ⫺5 Terminal side of Since sin ⫽ b/r ⫽ ⫺45, 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. 5-4 Trigonometric Functions 369 Use the Pythagorean theorem to find a: a2 ⫹ (⫺4)2 ⫽ 52 a2 ⫽ 9 a ⫽ ⫾3 ⫽3 a cannot be negative because is a IV quadrant angle. Using (a, b) ⫽ (3, ⫺4) and r ⫽ 5, we have MATCHED PROBLEM 4 cos ⫽ a 3 ⫽ r 5 tan ⫽ b ⫺4 4 ⫽ ⫽⫺ a 3 3 sec ⫽ r 5 ⫽ a 3 cot ⫽ csc ⫽ r 5 5 ⫽ ⫽⫺ b ⫺4 4 a 3 3 ⫽ ⫽⫺ b ⫺4 4 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 ⫽ ⫺34. Exact Values for Special Angles and Real Numbers Assuming a trigonometric function is defined, it can be evaluated exactly without 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. FIGURE 5 b Quadrantal angles. (0, 1) (⫺1, 0) (1, 0) a In each case, r ⫽ 兹a2 ⫹ b2 ⫽ 1, a positive number. (0, ⫺1) EXAMPLE 5 Trig Functions of Quadrantal Angles Find (A) sin 90° (B) cos (C) tan (⫺2) (D) cot (⫺180°) 370 5 TRIGONOMETRIC FUNCTIONS Solutions 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. b b 1 ⫽ ⫽1 (A) sin 90° ⫽ r 1 (a, b) ⫽ (0, 1), r ⫽ 1 a ⫺1 ⫽ ⫺1 ⫽ (B) cos ⫽ r 1 b 0 ⫽ ⫽0 (C) tan (⫺2) ⫽ a 1 (D) cot (⫺180°) ⫽ a b (a, b) ⫽ (⫺1, 0), r ⫽ 1 a b (a, b) ⫽ (1, 0), r ⫽ 1 a b a ⫺1 ⫽ b 0 (a, b) ⫽ (⫺1, 0), r ⫽ 1 a Not defined MATCHED PROBLEM 5 Explore/Discuss 2 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. b a a ␣ b P (a, b) (a, b) ⫽ (0, 0) ␣ is always positive 5-4 Trigonometric Functions 371 Figure 6 shows several reference triangles and reference angles corresponding to particular angles. FIGURE 6 Reference triangle Reference triangles and reference angles. ␣ Reference angle Reference angle ⫽ ⫺45⬚ ⫽ 120⬚ ␣ 180⬚ Reference triangle ⫺90⬚ (a) ␣ ⫽ 180° ⫺ 120° ⫽ 60° (b) ␣ ⫽ ⫽ 45° ⱍⱍ /2 ⫽ 5/4 ␣ ⫽ /6 ⫽ ␣ (c) 5 ␣⫽ ⫺⫽ 4 4 (d) ␣⫽⫽ ␣ 180⬚ 360⬚ ⫽ 420⬚ ␣ ⫺ ⫽ ⫺7/6 (e) ␣ ⫽ 420° ⫺ 360° ⫽ 60° 6 ␣⫽ (f) 7 ⫺⫽ 6 6 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 terminal 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: 372 5 TRIGONOMETRIC FUNCTIONS c ⫽ 2a FIGURE 7 30°–60° right triangle. b ⫽ 兹c2 ⫺ a2 30⬚ 30⬚ c ⫽ 兹(2a)2 ⫺ a2 c ⫽ 兹3a2 ⫽ a兹3 b 60⬚ a兹3 60⬚ (/3) 60⬚ a 30⬚ (/6) 2a a a c Similarly, using the Pythagorean theorem on a 45° right triangle, we obtain the result shown in Figure 8. c ⫽ 兹a2 ⫹ a2 FIGURE 8 45° right triangle. ⫽ 兹2a2 45⬚ c a兹2 ⫽ a兹2 a 45⬚ (/4) a 45⬚ (/4) a 45⬚ a 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 兹2. FIGURE 9 30°–60° AND 45° SPECIAL TRIANGLES 30⬚ (/6) 2 兹3 45⬚ (/4) 兹2 60⬚ (/3) 1 1 45⬚ (/4) 1 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- 5-4 Trigonometric Functions 373 nal side of the angle. Using the definition of the trigonometric functions, Definition 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. EXAMPLE 6 Solutions FIGURE 10 Exact Evaluation Evaluate exactly using appropriate reference triangles. (A) cos 60°, sin (/3), tan (/3) (B) sin 45°, cot (/4), sec (/4) (A) Use the special 30°–60° triangle with sides 1, 2, and 兹3 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 appropriate definitions. b (a, b) ⫽ (1, 兹3) r⫽2 2 cos 60° ⫽ sin b 兹3 ⫽ ⫽ 3 r 2 tan b 兹3 ⫽ ⫽ ⫽ 兹3 3 a 1 兹3 60⬚ (/3) a 1 a 1 ⫽ r 2 (B) Use the special 45° triangle with sides 1, 1, and 兹2 as the reference triangle, 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. FIGURE 11 b (a, b) ⫽ (1, 1) r ⫽ 兹2 兹2 1 6 a 1 b 兹2 ⫽ or r 兹2 2 cot a 1 ⫽ ⫽ ⫽1 4 b 1 sec r 兹2 ⫽ ⫽ ⫽ 兹2 4 a 1 1 45⬚ (/4) MATCHED PROBLEM sin 45° ⫽ Evaluate exactly using appropriate reference triangles. (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 multiples of /3 (60°), /6 (30°), and /4 (45°). These are illustrated in Figure 12. 374 5 TRIGONOMETRIC FUNCTIONS 3 6 2 3 3 4 6 ⫽ 2 2 3 ⫽ 2 4 2 6 ⫽ ⫽ 6 3 0 ⫽ 2 6 6 ⫽ 12 6 7 6 4 3 8 6 5 3 4 3 10 6 9 6 (a) Multiples of /3 (60⬚) 0 ⫽ 2 4 4 ⫽ ⫽ 4 ⫽ 11 6 ⫽ 2 3 4 6 5 6 3 3 3 ⫽ 8 4 5 4 5 3 0 ⫽ 2 7 4 6 4 3 2 ⫽ 3 2 (c) Multiples of /4 (45⬚) (b) Multiples of /6 (30⬚) FIGURE 12 Multiples of special angles. EXAMPLE 7 Solutions Exact Evaluation Evaluate exactly using appropriate reference triangles. (A) cos (7/4) (B) sin (2/3) (C) tan 210° (D) sec (⫺240°) 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) cos 7 1 兹2 ⫽ or 4 2 兹2 (B) sin 2 兹3 ⫽ 3 2 b b 7 4 1 a 4 (a, b) ⫽ (⫺1, 兹3) r⫽2 2 兹3 ⫺1 兹2 (a, b) ⫽ (1, ⫺1) r ⫽ 兹2 (C) tan 210° ⫽ ⫺1 1 兹3 ⫽ or ⫺ 兹3 兹3 3 210⬚ ⫺兹3 30⬚ ⫺1 2 (a, b) ⫽ (⫺兹3, ⫺1) r⫽2 a ⫺1 (D) sec (⫺240°) ⫽ (a, b) ⫽ (⫺1, 兹3) r⫽2 b 2 3 3 2 ⫽ ⫺2 ⫺1 b a 兹3 2 60⬚ ⫺1 a ⫺240⬚ 5-4 Trigonometric Functions MATCHED PROBLEM 7 Evaluate exactly using appropriate triangles. (A) tan (⫺/4) (B) sin 210° (C) cos (2/3) 375 (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 EXAMPLE 8 Solutions Find the smallest positive in degree and radian measure for which each is true. (A) tan ⫽ 1/兹3 (B) sec ⫽ ⫺ 兹2 b 1 ⫽ a 兹3 We can let (a, b) ⫽ (兹3, 1) or (⫺兹3, ⫺1). The smallest positive for which this is true is a quadrant I angle with reference triangle as drawn in Figure 13. (A) tan ⫽ FIGURE 13 b ⫽ 30° or 6 (兹3, 1) 1 30⬚ a 兹3 r 兹2 ⫽ Because r ⬎ 0 a ⫺1 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. (B) sec ⫽ FIGURE 14 b ⫽ 135° or 兹2 1 45⬚ ⫺1 a 3 4 376 5 TRIGONOMETRIC FUNCTIONS MATCHED PROBLEM 8 Remark Find the smallest positive in degree and radian measure for which each is true. (A) sin ⫽ 兹3/2 (B) cos ⫽ ⫺1/兹2 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. T A B L E 1 Special Angle Values sin 0° 30° 45° 60° 90° 0 1 2 1/兹2 or 兹2/2 兹3/2 1 cos tan 1 兹3/2 1/兹2 or 兹2/2 0 1/兹3 or 兹3/3 1 兹3 Not defined 1 2 0 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. Explore/Discuss 3 Fill in the cosine column in Table 2 with a pattern of values that is similar to those in the sine column. Discuss how the two columns of values are related. T A B L E —Memory Aid 2 Special Angle Values— sin 0° 30° 45° 60° 90° 兹0/2 ⫽ 0 兹1/2 ⫽ 12 兹2/2 兹3/2 兹4/2 ⫽ 1 cos 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. 5-4 Trigonometric Functions 377 Answers to Matched Problems 1. 2. 3. 4. 5. 6. 7. 8. (A) ⫺1 (B) ⫺ 12 (C) 1 (D) ⫺2 (A) ⫺0.8582 (B) 0.7539 (C) 1.542 (D) ⫺0.6383 (E) ⫺4.277 (F) 1.137 sin ⫽ ⫺ 45, cos ⫽ ⫺ 35, tan ⫽ 43, csc ⫽ ⫺ 54, sec ⫽ ⫺ 53, cot ⫽ 34 sin ⫽ 35, cos ⫽ ⫺ 45, csc ⫽ 53, sec ⫽ ⫺ 54, cot ⫽ ⫺ 43 (A) ⫺1 (B) ⫺1 (C) Not defined (D) 0 (A) cos 45° ⫽ 1/兹2, tan (/4) ⫽ 1, csc (/4) ⫽ 兹2 (B) sin 30° ⫽ 12, cos (/6) ⫽ 兹3/2, cot (/6) ⫽ 兹3 1 1 (A) ⫺1 (B) ⫺ 2 (C) ⫺ 2 (D) 2/兹3 (A) 60° or /3 (B) 135° or 3/4 EXERCISE 5-4 B A In Problems 33–48, evaluate exactly, using reference angles where appropriate, without using a calculator. 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. 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) 48. tan 690° 1. (6, 8) 2. (⫺3, 4) 3. (⫺1, 兹3) 4. (兹3, 1) 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° 47. cos 510° 7. cot 12 8. csc 13 10. tan 4.327 For which values of 0° ⱕ ⬍ 360°, is each of Problems 49–54 not defined? Explain why. 11. cot (⫺431.41°) 12. sec (⫺247.39°) 49. cos 50. sec 51. tan 13. sin 113°27⬘13⬙ 14. cos 235°12⬘47⬙ 52. cot 53. csc 54. sin 9. sin 2.137 In Problems 15–26, evaluate exactly, using reference triangles where appropriate, without using a calculator. In Problems 55–60, find the smallest positive in degree and radian measure for which 15. sin 0° 16. cos 0° 17. tan 60° 55. cos ⫽ 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° 30. ⫽ 4 28. ⫽ 135° 31. ⫽ ⫺ 5 3 29. ⫽ 7 6 32. ⫽ ⫺ 56. sin ⫽ ⫺1 ⫺兹3 57. sin ⫽ 2 2 58. tan ⫽ ⫺兹3 59. csc ⫽ Find the reference angle ␣ for each angle in Problems 27–32. 27. ⫽ 300° ⫺1 2 5 4 ⫺2 兹3 60. sec ⫽ ⫺兹2 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. sin ⫽ 35 62. tan ⫽ ⫺ 43 and and 63. cos ⫽ ⫺兹5/3 cos ⬍ 0 sin ⬍ 0 and cot ⬎ 0 378 5 TRIGONOMETRIC FUNCTIONS 64. cos ⫽ ⫺兹5/3 and tan ⬎ 0 65. Which trigonometric functions are not defined when the terminal side of an angle lies along the vertical axis. Why? Find light intensity I in terms of k for ⫽ 0°, ⫽ 30°, and ⫽ 60°. 66. Which trigonometric functions are not defined when the terminal side of an angle lies along the horizontal axis? Why? Sun 67. Find exactly, all , 0° ⱕ ⬍ 360°, for which cos ⫽ ⫺兹3/2. 68. Find exactly, all , 0° ⱕ ⬍ 360°, for which cot ⫽ ⫺1/兹3. Solar cell 69. Find exactly, all , 0 ⱕ ⬍ 2, for which tan ⫽ 1. 70. Find exactly, all , 0 ⱕ ⬍ 2, for which sec ⫽ ⫺兹2. C —Engineering. The figure illustrates a piston 77. Physics— connected to a wheel that turns 3 revolutions per second; hence, the angle is being generated at 3(2) ⫽ 6 radians per second, or ⫽ 6t, where t is time in seconds. If P is at (1, 0) when t ⫽ 0, show that s P (a, b) 76. Solar Energy. Refer to Problem 75. Find light intensity I in terms of k for ⫽ 20°, ⫽ 50°, and ⫽ 90°. A y ⫽ b ⫹ 兹42 ⫺ a2 ⫽ sin 6t ⫹ 兹16 ⫺ (cos 6t)2 for t ⱖ 0. y 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 y 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 (6兹3, 6). What is the length of the arc on the circle in quadrant I between the positive horizontal axis and the point (6兹3, 6)? 74. In a rectangular coordinate system, a circle with center at the origin passes through the point (2, 2兹3). What is the length of the arc on the circle in quadrant I between the positive horizontal axis and the point (2, 2兹3)? 4 inches 3 revolutions per second a P (a, b) b (1, 0) x ⫽ 6t APPLICATIONS 75. Solar Energy. The intensity of light I on a solar cell changes with the angle of the sun and is given by the formula I ⫽ k cos , where k is a constant (see the figure). —Engineering. In Problem 77, find the position 78. Physics— of the piston y when t ⫽ 0.2 second (to three significant digits). 5-5 Solving Right Triangles r⫽1 379 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°. n⫽8 ★ 79. Geometry. The area of a rectangular n-sided polygon circumscribed about a circle of radius 1 is given by A ⫽ n tan (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. 180° n y (A) Find A for n ⫽ 8, n ⫽ 100, n ⫽ 1,000, and n ⫽ 10,000. Compute each to five decimal places. L L (B) What number does A seem to approach as n → ⬁? (What is the area of a circle with radius 1?) ★ x 80. Geometry. The area of a regular n-sided polygon inscribed in a circle of radius 1 is given by A⫽ 360° n sin 2 n (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 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. Section 5-5 Solving Right Triangles* FIGURE 1 ␣ c  a b 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. Trigonometric functions play a central role in this process. To start, we locate a right triangle in the first quadrant of a rectangular coordinate system and observe, from the definition of the trigonometric functions, six trigonometric ratios involving the sides of the triangle. [Note that the right triangle is the reference triangle for the angle .] *This section provides a significant application of trigonometric functions to real-world problems. However, 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.
