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5.2 Sum and Difference Formulas Objectives Isten to the Same Note Played on a Piano and a ᕡ Use the Formula for the Violin

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5.2 Sum and Difference Formulas Objectives Isten to the Same Note Played on a Piano and a ᕡ Use the Formula for the Violin P-BLTZMC05_585-642-hr 21-11-2008 12:53 Page 596 596 Chapter 5 Analytic Trigonometry Critical Thinking Exercises Group Exercise Make Sense? In Exercises 88–91, determine whether each 97. Group members are to write a helpful list of items for a statement makes sense or does not make sense, and explain pamphlet called “The Underground Guide to Verifying your reasoning. Identities.” The pamphlet will be used primarily by students who sit, stare, and freak out every time they are asked to 88. The word identity is used in different ways in additive verify an identity. List easy ways to remember the fundamen- identity, multiplicative identity, and trigonometric identity. tal identities. What helpful guidelines can you offer from the 89. To prove a trigonometric identity, I select one side of the perspective of a student that you probably won’t find in math equation and transform it until it is the other side of the books? If you have your own strategies that work particularly equation, or I manipulate both sides to a common trigono- well, include them in the pamphlet. metric expression. cos x sin x Preview Exercises 90. In order to simplify - , I need to know how 1 - sin x cos x Exercises 98–100 will help you prepare for the material covered in to subtract rational expressions with unlike denominators. the next section. 91. The most efficient way that I can simplify 98. Give exact values for cos 30°, sin 30°, cos 60°, sin 60°, sec x + 1 sec x - 1 cos 90°, and sin 90°. 1 21 2 2 is to immediately rewrite the sin x 99. Use the appropriate values from Exercise 98 to answer each expression in terms of cosines and sines. of the following. a. Is cos 30° + 60° , or cos 90°, equal to cos 30° + cos 60°? In Exercises 92–95, verify each identity. 1 2 b. Is cos 30° + 60° , or cos 90°, equal to sin3 x - cos3 x 1 -2 92. = 1 + sin x cos x cos 30° cos 60° sin 30° sin 60°? sin x - cos x sin x - cos x + 1 sin x + 1 100. Use the appropriate values from Exercise 98 to answer each 93. = of the following. sin x + cos x - 1 cos x tan2 x-sec2 x a. Is sin 30° + 60° , or sin 90°, equal to sin 30° + sin 60°? 94. ln ƒ sec x ƒ =-ln ƒ cos x ƒ 95. ln e =-1 1 2 b. Is sin 30° + 60° , or sin 90°, equal to 96. Use one of the fundamental identities in the box on page 586 1 2 sin 30° cos 60° + cos 30° sin 60°? to create an original identity. Section 5.2 Sum and Difference Formulas Objectives isten to the same note played on a piano and a ᕡ Use the formula for the violin. The notes have a different quality or cosine of the difference of L “tone.” Tone depends on the way an instrument two angles. vibrates. However, the less than 1% of the popula- ᕢ Use sum and difference tion with amusia, or true tone deafness, cannot tell formulas for cosines and sines. the two sounds apart. Even simple, ᕣ Use sum and difference familiar tunes such as Happy Birthday formulas for tangents. and Jingle Bells are mystifying to amusics. When a note is played, it vibrates at a specific fundamental frequency and has a particular amplitude. Amusics cannot tell the difference between two sounds from tuning forks modeled by p = 3 sin 2t and p = 2 sin 2t + p , respectively. 1 2 However, they can recognize the difference between the two equations. Notice that the second equation contains the sine of the sum of two angles. In this section, we will be developing identities involving the sums or differences of two angles. These formulas are called the sum and difference formulas. We begin with cos a - b , the cosine of 1 2 the difference of two angles. P-BLTZMC05_585-642-hr 21-11-2008 12:53 Page 597 Section 5.2 Sum and Difference Formulas 597 y The Cosine of the Difference of Two Angles Q = (cos α, sin α) P = (cos β, sin β) The Cosine of the Difference of Two Angles a a - b = a b + a b b cos cos cos sin sin x 1 2 a − b The cosine of the difference of two angles equals the cosine of the first angle times the cosine of the second angle plus the sine of the first angle times the sine of the second angle. x2 + y2 = 1 We use Figure 5.1 to prove the identity in the box. The graph in Figure 5.1(a) (a) shows a unit circle, x2 + y2 = 1. The figure uses the definitions of the cosine and sine functions as the x- and y-coordinates of points along the unit circle. For exam- y ple, point P corresponds to angle b. By definition, the x-coordinate of P is cos b and the y-coordinate is sin b. Similarly, point Q corresponds to angle a. By Q = (cos (α − β), sin (α − β)) definition, the x-coordinate of Q is cos a and the y-coordinate is sin a. Note that if we draw a line segment between points P and Q, a triangle is P = (1, 0) formed.Angle a - b is one of the angles of this triangle.What happens if we rotate x this triangle so that point P falls on the x-axis at (1, 0)? The result is shown in Figure 5.1(b). This rotation changes the coordinates of points P and Q. However, it a − b has no effect on the length of line segment PQ. We can use the distance formula, d = 4 x - x 2 + y - y 2 , to find an 1 2 12 1 2 12 x2 + y2 = 1 expression for PQ in Figure 5.1(a) and in Figure 5.1(b). By equating the two expressions for PQ, we will obtain the identity for the cosine of the difference of (b) two angles,a - b. We first apply the distance formula in Figure 5.1(a). Figure 5.1 Using the unit circle andPQ to develop a formula for cos a - b 1 2 PQ = 4 cos a - cos b 2 + sin a - sin b 2 Apply the distance formula, 1 2 1 2 d = 4 x - x 2 + y - y 2 , to 1 2 12 1 2 12 find the distance between cos b, sin b and cos a, sin a . 1 2 1 2 = 3cos2 a - 2 cos a cos b + cos2 b + sin2 a - 2 sin a sin b + sin2 b Square each expression using A - B 2 = A2 - 2AB + B2. 1 2 = 4 sin2 a + cos2 a + sin2 b + cos2 b - 2 cos a cos b - 2 sin a sin b Regroup terms to apply a Pythagorean 1 2 1 2 identity. = 21 + 1 - 2 cos a cos b - 2 sin a sin b Because sin2 x + cos2 x = 1, each expression in parentheses equals 1. = 22 - 2 cos a cos b - 2 sin a sin b Simplify. Next, we apply the distance formula in Figure 5.1(b) to obtain a second expression for PQ. We let x , y = 1, 0 and x , y = cos a - b , sin a - b . 1 1 12 1 2 1 2 22 1 1 2 1 22 PQ = 4 cos a - b - 1 2 + sin a - b - 0 2 Apply the distance formula to find the distance between 3 1 2 4 3 1 2 4 (1, 0) and cos a - b , sin a - b . 1 1 2 1 22 = 4cos2 a - b - 2 cos a - b + 1 + sin2 a - b Square each expression. 1 2 1 2 1 2 =͙cos2 (a-b)-2 cos (a-b)+1+sin2 (a-b) Using a Pythagorean identity, sin2 (a − b) + cos2 (a − b) = 1. ͙ = 1-2 cos (a-b)+1 Use a Pythagorean identity. = 42 - 2 cos a - b Simplify. 1 2 P-BLTZMC05_585-642-hr 21-11-2008 12:53 Page 598 598 Chapter 5 Analytic Trigonometry Now we equate the two expressions for PQ. 4 2 - 2 cos a - b = 22 - 2 cos a cos b - 2 sin a sin b The rotation does not 1 2 change the length of PQ. 2 - 2 cos a - b = 2 - 2 cos a cos b - 2 sin a sin b Square both sides to 1 2 eliminate radicals. -2 cos a - b =-2 cos a cos b - 2 sin a sin b Subtract 2 from both 1 2 sides of the equation. cos a - b = cos a cos b + sin a sin b Divide both sides of 1 2 - ᕡ Use the formula for the cosine of the equation by 2. the difference of two angles. This proves the identity for the cosine of the difference of two angles. Now that we see where the identity for the cosine of the difference of two Sound Quality angles comes from, let’s look at some applications of this result. and Amusia EXAMPLE 1 Using the Difference Formula People with true tone deafness for Cosines to Find an Exact Value cannot hear the difference among Find the exact value of cos 15°. tones produced by a tuning fork, a flute, an oboe, and a violin. They Solution We know exact values for trigonometric functions of 60° and 45°. Thus, cannot dance or tell the difference - between harmony and dissonance. we write 15° as 60° 45° and use the difference formula for cosines.
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