514 CHAPTER 7 Analytic Trigonometry Now Work PROBLEMS 27 AND 31 In Words Formula (4) states that the sine of the sum of two angles equals the sine of the first angle times the cosine of the second angle plus the cosine of the first angle times the sine of the second angle. Using the Sum Formula to Find an Exact Value Find the exact value of π sin 7 12 . Solution EXAMPLE 3 π π π π π π π π π ( ) ( ) ( ) = + = + = + = ⋅ + ⋅ = + sin 7 12 sin 3 12 4 12 sin 4 3 sin 4 cos 3 cos 4 sin 3 2 2 1 2 2 2 3 2 1 4 2 6 Sum Formula for sine Using the Difference Formula to Find an Exact Value Find the exact value of ° °− ° ° sin80 cos20 cos80 sin20 . Solution EXAMPLE 4 The form of the expression ° °− ° ° sin80 cos20 cos80 sin20 is that of the right side of formula (5) for α β ( ) − sin with α = ° 80 and β = ° 20 . That is, ( ) ° °− ° ° = °− ° = ° = sin80 cos20 cos80 sin20 sin 80 20 sin60 3 2 Finding Exact Values If α π α π = < < sin 4 5 , 2 , and β = − = − sin 2 5 2 5 5 , π β π < < 3 2 , find the exact value of each of the following. (a) α cos (b) β cos (c) α β ( ) + cos (d) α β ( ) + sin EXAMPLE 5 Proof sin cos 2 cos 2 cos 2 cos sin 2 sin sin cos cos sin sin sin sin cos cos sin sin cos cos sin sin cos cos sin α β π α β π α β π α β π α β α β α β α β α β α β α β α β α β α β α β ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) + = ⎡ − + ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − − ⎣ ⎢ ⎤ ⎦ ⎥ = − + − = + − = + − = − + − = + − = − Identity (3a) Difference Formula for cosine Identities (3a) and (3b) Use the Sum Formula for sine just obtained. Even—Odd Identities ■ THEOREM Sum and Difference Formulas for the Sine Function • α β α β α β ( ) + = + sin sin cos cos sin (4) • α β α β α β ( ) − = − sin sin cos cos sin (5) Now Work PROBLEM 21
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