512 CHAPTER 7 Analytic Trigonometry Figure 32 1 O 21 a 2 b 1 21 y x P3 5 (cos( a 2 b), sin( a 2 b)) x2 1 y2 5 1 x2 1 y2 5 1 A 5 (1, 0) (b) 1 O 21 a a 2 b b 1 21 y x P2 5 (cos a, sin a) P1 5 (cos b, sin b) (a) Now place the angle α β − in standard position, as shown in Figure 32(b). The point A has coordinates ( ) 1, 0 , and the point P3 is on the terminal side of the angle α β − , so its coordinates are cos , sin . α β α β ( ) ( ) ( ) − − Looking at triangle OPP1 2 in Figure 32(a) and triangle OAP3 in Figure 32(b), note that these triangles are congruent. (Do you see why? SAS: two sides and the included angle, α β − , are equal.) As a result, the unknown side of triangle OPP1 2 and the unknown side of triangle OAP3 must be equal; that is, ( ) ( ) = d A P d P P , , 3 1 2 Now use the distance formula to obtain α β α β α β α β α β α β α β α β α β α β α β α α β β α α β β α β α β α β α β α β α β α β α β α β ( ) [ ] ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − − + − − = − + − − − + − = − + − − − − + + − = − + + − + − − = − − − − =− − − = + cos 1 sin 0 cos cos sin sin cos 1 sin cos cos sin sin cos 2 cos 1 sin cos 2 cos cos cos sin 2 sin sin sin 2 2 cos 2 2 cos cos 2 sin sin 2 cos 2 cos cos 2 sin sin cos cos cos sin sin 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ) ( ) = d A P d P P , , 3 1 2 Square both sides. Multiply out the squared terms. Use a Pythagorean Identity (3 times). Subtract 2 from both sides. Divide both sides by −2. This is formula (2). The proof of the Sum Formula for cosine follows from the Difference Formula for cosine and the Even–Odd Identities. Because α β α β ( ) + = − − , it follows that α β α β α β α β α β α β ( ) ( ) [ ] ( ) ( ) + = − − = − + − = − cos cos cos cos sin sin cos cos sin sin ■ 1 Use Sum and Difference Formulas to Find Exact Values One use of the Sum and Difference Formulas is to obtain the exact value of the cosine of an angle that can be expressed as the sum or difference of angles whose sine and cosine are known exactly. Using the Sum Formula to Find an Exact Value Find the exact value of cos ° 75 . Solution EXAMPLE 1 Because ° = ° + ° 75 45 30 , use formula (1) to obtain Sum Formula for cosine ( ) ( ) ° = ° + ° = ° °− ° ° ↑ = ⋅ − ⋅ = − cos 75 cos 45 30 cos 45 cos 30 sin 45 sin30 2 2 3 2 2 2 1 2 1 4 6 2 Use the Difference Formula for cosine. Use the Even—Odd Identities.
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