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CHAPTER 8: TRIGONOMETRIC FUNCTIONS 157 by the term sin2 x, Also, if you divide both sides of you obtain the identity + = , which you can simplify to: 1 + cot2 x = csc2 x. This collection of related identities (1) sin2 x + cos2 x = 1 (2) tan2 x + 1 = sec2 x (3) 1 + cot2 x = csc2 x is very important in simplifying some complex trigonometric expressions. Be sure you can also recognize these identities when they appear in other forms: 1 – sin2 x = cos2 x 1 – cos2 x = sin2 x sec2 x – 1 = tan2 x csc2 x – 1 = cot2 x Other important trigonometric identities include the addition formulas: sin (A + B) = sin A cos B + sin B cos A cos (A + B) = cos A cos B – sin A sin B Since A – B = A + –B, you can also obtain sin (A – B) = sin A cos (–B) + sin (–B) cos A = sin A cos B – sin B cos A cos (A – B) = cos A cos (–B) – sin A sin (–B) = cos A cos B + sin A sin B Letting A = B in the addition formulas, you can obtain the identities: sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A double-angle From this last double-angle identity you can derive the half-angle identities: 2 2 These identities can also be written: sin2 x = (1 – cos x) cos2 x = (1 + cos x) 2 cos 2A = cos A – (1 – cos A) = 2 cos A – 1 or, cos2 A = (1 + cos 2A) cos 2A = (1 – sin2 A) – sin2 A = 1 – 2 sin2 A or, sin2 A = (1 – cos 2A) These are the identities that are most often applied in elementary analysis. They will help you to simplify more complex trigonometric expressions and to do some more difficult computations. 158 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Find sin . Since is not a multiple of or or that you should be familiar with, you would not have been able to find this before, without using your calculator. However, if you notice that = + sin = + = sin ( , you can rewrite + ) = sin = cos = + + sin . + a. cos b. sin c. cot cos + = Calculate: Ans.: a. ( b. ( c. ( = + ) = = ) = .9659 Simplify cos ( – x). cos ( – x) = cos cos (–x) – sin = (0) cos x – (1) (–sin x) = sin x To find tan tan , notice that = = = sin (–x) Simplify: a. cos ( – ) b. tan ( + ) . = Ans.: a. –cos b. –cot = = = .4142 = – ) CHAPTER 8: TRIGONOMETRIC FUNCTIONS 159 To obtain an identity for cos 3x, note 3x = 2x + x. cos 3x = cos (2x + x) = cos 2x cos x – sin 2x sin x = (cos2 x – sin2 x) cos x – 2 sin x cos x sin x = cos3 x – sin2 x cos x – 2 sin2 x cos x = cos3 x – 3 sin2 x cos x = cos3 x – 3 (1 – cos2 x) cos x = cos3 x – 3 cos x + 3 cos3 x = 4 cos 3 x – 3 cos x Right Triangle Trigonometry Using the definitions of the trigonometric functions with the relationships of the sides of a right triangle as described on page 148, you can calculate the value of any trigonometric function defined on the angles of such a triangle. If is the angle in the triangle for which you wish to calculate the trigonometric functions ( figure on page 148), sin is the angle placed at the center of the circle in the = tan sec , = cos , = cot , csc = = = From these identities, you can calculate the trigonometric functions for any angle by considering the angle to be placed in a right triangle at the center of a circle. The reference circle is not necessarily the unit circle, since the hypotenuse of the triangle is the radius. However, you can compute the trigonometric function values by using the lengths of the sides of the triangle. Write sin 3x using only the functions sin x and cos x. Ans.: 3 sin x cos2 x – sin3 x or 3 sin x – 4 sin3 x 160 CHAPTER 8: TRIGONOMETRIC FUNCTIONS For example, if the hypotenuse is 5 and the length of the side opposite the angle is 3, you can find the length of the third side (the side adjacent to ) by using the Pythagorean Theorem to obtain = = = 4. You can then calculate all the trigonometric functions of , without even knowing the value of : sin = cos = tan = cot = sec = csc = Suppose that instead of knowing the lengths of the sides, you know that tan = . The opposite and adjacent sides are in the ratio of 2 : 3, or l2 : 3l. You can use the Pythagorean Theorem to find the hypotenuse, = = = . Since the values of the trigonometric functions depend on the ratio of the sides, use 2 for the opposite side, 3 for the adjacent side and for the hypotenuse. The values of the trigonometric functions are: If sin sin = cos = tan = cot = sec = csc = = , then using x for the length of the opposite side and 2 for the hypotenuse, the length of the adjacent side is = . The trigonometric functions are: sin = cos = tan = cot = sec = cot = CHAPTER 8: TRIGONOMETRIC FUNCTIONS 161 The measurement unit used for angles is degree measure. When you want to obtain the trigonometric functions for an angle of known degree measure, move the angle to the center of the unit circle and look at the ratios of the sides. The central angle corresponding to an arc of length around the unit Use the diagrams on page 140. circle is 30°. To find the trigonometric functions for this angle, use the ratio of the sides of the right triangle with a 30° angle: sin 30° = cos 30° = tan 30° = cot 30° = sec 30° = csc 30° = 2 Angles that fit in the first quadrant are called acute angles. For larger angles you use the unit circle to get the signs right the same way you do finding the trigonometric functions for arcs. You can do this for angles measured either in the positive (counterclockwise) or negative (counterclockwise) directions. To find the trigonometric functions for an angle of 135°, fit the angle into the unit circle. You can see that two angles are formed in the first two quadrants, the 135° angle, and the supplementary 45° angle that corresponds to the arc from the end of the 135° angle to the point (–1, 0). Then calculate the trigonometric functions using the supplementary 45° angle that lies in the second quadrant. sin 135° = cos 135° = tan 135° = –1 cot 135° = –1 sec 135° = csc 135° = You can also use your calculator to find the value of the trigonometric functions for angles other than integer multiples of 30° and 45°. Just be sure that you have the calculator set in the appropriate mode. 1 162 CHAPTER 8: TRIGONOMETRIC FUNCTIONS To find sin 54°, put your calculator in degree mode and press the key, then 54 and . The calculator finds sin 54° = .8090. Find: a. cos 138° b. cot 53° c. sec To get sec 54°, since there is no key, you find cos 54° by using the key. You have cos 54° = .5878. Then use the reciprocal key, and to find To find tan , make sure you are back in radian mode and use the key, followed by , = 1.7013 = sec 54°. (the 2nd function for the ^ key), ÷ , 15, and . When you press To obtain csc and , find sin to obtain you get tan = .2126. (.2079), then use the reciprocal key = csc = 4.8097. Applications Whenever you know the lengths of two sides of a right triangle you can find the length of the third side using the Pythagorean Theorem and then the values of all the trigonometric functions. If you have one of the acute angles and the length of one side you can find all the remaining parts of the triangle since you can find the other acute angle and then use the trigonometric functions to find the other sides. If you know that one angle of a right triangle is 26°, and the length of the side opposite this angle is 11, you can find the hypotenuse from the equation sin 26° = . To find the length of the hypotenuse, (abbreviated to h here), h sin 26° = 11 h= h= = 25.09 Ans.: a. –.7431 b. .7536 c. 1.0642 CHAPTER 8: TRIGONOMETRIC FUNCTIONS 163 You can find the adjacent side either by using the Pythagorean Theorem, = = 22.55, or, tan 26° = = a tan 26° = 11 a= = = 22.55 You can apply this method for solving right triangles in many situations that require you to find the lengths of parts of right triangles. When you set the base of a ladder 5 feet from the base of a flat wall, the ladder makes an angle of 48° with the ground. How high is the top of the ladder? The diagram on the right shows the ladder leaning against the wall, which forms the side of the triangle opposite the 48° angle. The part of the triangle you wish to find is the side opposite the angle. You already know the adjacent side, so the trigonometric function for you to use is the tangent, since the tangent is the ratio of the lengths of the opposite and adjacent sides. How high would the ladder reach if the angle with the ground was 73° (and the bottom of the ladder was still 5 feet from the base of the wall? tan 48° = 1.1106 = (1.1106)(5) = x Ans.: 16.35 feet x = 5.55 feet 164 CHAPTER 8: TRIGONOMETRIC FUNCTIONS The ladder reaches 5.55 feet up the wall. Since the ladder is 5 feet from the base of the wall, you can use the Pythagorean Theorem to find the length of the ladder. = = = 7.47 feet You are standing on the top of a cliff so that your eye level is 100 feet above the surface of a lake, looking through a telescope at a boat that is approaching the cliff. To keep the image of the boat in focus, you have to drop your telescope 20° from the horizontal position. How far away from the cliff is the boat? The diagram on the right shows the line of sight of the telescope dropped 20° from the horizontal line of sight to the boat. Therefore the complementary angle inside the triangle is 70°. You know that the side adjacent to the 70° angle is 100 feet long, and the side opposite this angle is the quantity you are looking for. The trigonometric function to use is again the tangent. tan 70° = 2.7475 = (2.7475)(100) = x x = approximately 275 feet CHAPTER 8: TRIGONOMETRIC FUNCTIONS Suppose you walk due north at 4 miles per hour and continue for 3 hours. Then you turn eastward 42° and continue at 3 miles per hour for 2 more hours before you stop. How far are you from your starting point? The diagram on the right is oriented so that north is up. Notice that the distance labeled y is opposite the 42° angle in the top right triangle, and the side labeled z is adjacent to the 42° angle. The hypotenuse of this triangle is 6 miles since you traveled for 2 hours at 3 miles per hour during the second part of the trip. The first (northern) leg of the trip was 3 hours long at 4 miles per hour and therefore covered a distance of 12 miles. You can find the distance labeled y by using the sine of the 42° angle. sin 42° = .6691 = (.6691)(6) = y y = 4.01 miles You can find the distance labeled z by using the cosine of the 42° angle. cos 42° = .7431 = (.7431)(6) = z z = 4.46 miles Now you can find the straight-line distance, x, from start to end by using the Pythagorean Theorem: = = = 16.94 miles 165 166 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Exercises 1. Give the correct sign for each of the following: a. sin 3 b. cos 7 c. tan 9 d. sin –5 e. sec –4 f. tan 3.5 2. Change to degree measure (each is given in radians): a. b. c. d. e. 2 f. 3. Change to radian measure (each is given in degrees): a. 120° b. 540° c. 225° d. –150° e. 144° f. –444° 4. Find each of the following: a. sin b. cos f. g. sec cot c. sin h. cot d. tan i. 5. Sketch graphs for each of the following functions a. f(x) = 3 sin x b. f(x) = 2 tan c. f(x) = sec x x d. f(x) = 4 cos 3x e. f(x) = sin (2x – ) f. f(x) = 4 csc x – 3 g. f(x) = –2 cot (x + ) h. f(x) = –1 + cos ( x – i. f(x) = 5 cos (2x – 1) – 2 j. f(x) = –3 + 6 sec ) x 6. Tell why each of the following is true or false (without calculation): a. cos 2 > cos 3 b. sec (1 + ) = sec 1 c. tan 2 > sec 2 d. csc 9 is negative e. sin 1 > sin (sin 1) e. csc csc j. sin CHAPTER 8: TRIGONOMETRIC FUNCTIONS 7. Express each of the following in terms of sin x and cos x: a. tan 2x b. cos 4x c. sin 4x 167 d. sin 3x + cos 2x 8. Use the trigonometric identities to find exact values for: a. tan b. cos c. 2 sin d. sec e. cot 9. Find the lengths of all sides of the right triangle, given a. = and the opposite side is 3. b. = and the adjacent side is 5. c. = 23° and the hypotenuse is 7. d. sin = . e. tan = . f. cos = . g. tan = h. sin = . . 10. If the angle of inclination from the ground to the top of a building is 72° at a point that is 50 feet from the building, how high is the building? 11. A woman is standing on a dock at some height above the water. She is holding one end of a rope attached to the front end of a boat that is 8 feet from the dock. The rope is stretched taut and is making an angle of 58° with her body. How high above the water is the dock? 12. A lifeguard is on the edge of a beach looking straight out into the water. When he turns 65° to his right he sees a drowning child 100 yards away from him in the direction he is facing. How far from the shore is the child? 168 CHAPTER 8: TRIGONOMETRIC FUNCTIONS Answers 1. a. + b. + c. – d. + e. – f. + 2. a. 300° b. 315° c. 48° d. –13.37° e. 114.59° f. –42.97° 3. a. b. c. d. e. f. 4. a. b. c. d. –1 e. –1 f. g. h. i. j. 1 2 5. All the following graphs are in the same window: X: [–7, 7] Y: [–10, 10] with the x-axis tick marks (Xscl) set at and the y-axis tick marks (Yscl) at 1. (a) (b) (d) (c) (e) CHAPTER 8: TRIGONOMETRIC FUNCTIONS (f) (g) 169 (h) (i) (j) 6. a. Both 2 radians and 3 radians are in the second quadrant, but 3 radians is closer to . Since the cosine becomes more negative as you move through the second quadrant, cos 3 is more negative than cos 2. b. If sec (1 + ) = sec 1, then cos (1 + ) = cos 1. However, traveling 1 radian around the circle puts you in the first quadrant, while a distance 1 + radians around the circle leaves you in the second. The cosine function is positive in the first quadrant, but negative in the second. Therefore, sec 1 and sec (1 + ) have different signs and cannot be equal. The statement is false. c. sin 2 < 1, since 2 radians is in the second quadrant. If you divide both sides of this inequality by the negative number, cos 2, the direction of the inequality is reversed, so tan 2 = > = sec 2 d. 9 radians is less than 3 radians, so 9 radians is in the second quadrant. The sine function, and therefore, the cosecant function, is positive in the second quadrant. The statement is false. e. Since sin 1 < 1, sin 1 radians is not as far around the first quadrant as 1 radian is. Therefore, the sine at 1 radian must be greater than the sine at sin 1 radians. 170 CHAPTER 8: TRIGONOMETRIC FUNCTIONS b. 1 – 8 sin2 x + 8 sin4 x 7. a. c. 4 sin x cos x – 8 sin3 x cos x d. 1 + 3 sin x – 2 sin2 x – 4 sin3 x 8. a. b. c. d. 9. a. hypotenuse = 6, adjacent side = 3 b. hypotenuse = 5 , opposite side = 5 c. opposite side = 7 sin 23° = 2.735, adjacent side = 7 cos 23° = 6.44 d. opposite side = 2, hypotenuse = 3, adjacent side = e. opposite side = 7, adjacent side = 3, hypotenuse = f. adjacent side = x, hypotenuse = 5, opposite side = g. opposite side = x, adjacent side = 1 + x, hypotenuse = h. opposite side = 1, hypotenuse = 10. 11. 12. 50 tan 72° = 153.88 feet = 5 feet 100 sin 25° = 42.26 yards , adjacent side = x e.