SECTION 7.5 Sum and Difference Formulas 513 Using the Difference Formula to Find an Exact Value Find the exact value of π cos 12 . EXAMPLE 2 Solution π π π π π π π π π ( ) ( ) ( ) = − = − = + = ⋅ + ⋅ = + cos 12 cos 3 12 2 12 cos 4 6 cos 4 cos 6 sin 4 sin 6 2 2 3 2 2 2 1 2 1 4 6 2 Use the Difference Formula for cosine. Now Work PROBLEMS 13 AND 19 Another use of the Sum and Difference Formulas is to establish other identities.Two important identities, conjectured in Section 6.4, are given next. • π θ θ ( ) − = cos 2 sin (3a) • π θ θ ( ) − = sin 2 cos (3b) Proof To establish identity (3a), use the Difference Formula for α β ( ) − cos with α π = 2 and β θ = . π θ π θ π θ θ θ θ ( ) − = + = ⋅ + ⋅ = cos 2 cos 2 cos sin 2 sin 0 cos 1 sin sin To establish identity (3b), use the identity (3a) just established. π θ π π θ θ ( ) ( ) − = ⎡ − − ⎣ ⎢ ⎤ ⎦ ⎥ = sin 2 cos 2 2 cos ↑ Use Identity (3a) ■ Seeing the Concept Graph Y x cos 2 1 π( ) = − and Y x sin 2 = on the same screen. Does doing this provide evidence of result (3a)? How would you provide evidence of the result (3b)? Also, because the cosine function is even π θ θ π θ π ( ) ( ) ( ) − = ⎡ − − ⎣ ⎢ ⎤ ⎦ ⎥ = − cos 2 cos 2 cos 2 ↑ Even Property of Cosine and because π θ θ ( ) − = cos 2 sin ↑ Identity (3a) it follows that θ π θ ( ) − = cos 2 sin . This means the graphs of θ π ( ) = − y cos 2 and θ = y sin are identical. Having established the identities (3a) and (3b), we now can derive the sum and difference formulas for α β ( ) + sin and α β ( ) − sin .
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