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(a) Write the product 2 sin(2π/9) cos(π/9) as a sum of function values.(b) Express cos55°cos45° as a sum.(c) Express the product sin6θ cos4θ as a sum.(d) Write the sum sin(π/9)+sin(2π/9) as a product of function values.(e) Express sin55°+sin45° as a product.(f) Use the product-to-sum to evaluate cos3π/2 cosπ/2.(g) Use the sum-to-product to evaluate cos3π/2+cosπ/2. |
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Answer» (a) 2 sin \(\frac {2\pi}{9}\) cos \(\frac {\pi}{9}\) = sin (\(\frac {2\pi}{9}\) + \(\frac {\pi}{9}\)) + sin (\(\frac {2\pi}{9}\) - \(\frac {\pi}{9}\)) = sin (\(\frac {3\pi}{9}\)) + sin (\(\frac {\pi}{9}\)) = sin (\(\frac {\pi}{9}\)) + sin (\(\frac {\pi}{9}\)) (b) cos 55° cos 45° = \(\frac 12 \) (cos (55° + 45°) + cos (55° - 45°)) = \(\frac 12 \) (cos 100° + cos 10°) (c) sin 6 θ cos 4 θ = \(\frac 12 \) (sin 6 θ + 4 θ) + sin ( 6 θ - 4 θ) = \(\frac 12 \) (sin 10 θ + 2 θ) (d) sin \(\frac {\pi}{9}\) + sin \(\frac {2\pi}{9}\) = 2 sin \((\frac {\frac {\pi}{9} + \frac {2\pi}{9}}{2}) cos (\frac {\frac \pi9 -\frac {2\pi}{9}}{2})\) = = 2 sin \((\frac {3\pi}{18}) cos ( \frac {-\pi}{18})\) (e) sin 55° + sin 45° = 2 sin \((\frac {55°+45°}{2}) cos (\frac {55°-45°}{2})\) = 2 sin \((\frac {100°}{2}) cos (\frac {10°}{2})\) = 2 sin 50° cos 5° (f) cos \(\frac {3\pi}{2} cos \frac {\pi}{2} \) = 0 x 0 = 0 (∵ \(cos \frac {\pi}{2} = \) 0 & cos \(\frac {3\pi}{2}\) = 0) (g) cos \(\frac {3\pi}{2} + cos \frac {\pi}{2} \) = 0 + 0 = 0 or (f) cos \(\frac {3\pi}{2} cos \frac {\pi}{2} \) = \(\frac 12 \) ( cos ( \(\frac {3\pi}{2} cos \frac {\pi}{2} \)) + cos ( \(\frac {3\pi}{2} - \frac {\pi}{2} \) )) = \(\frac 12 \) ( cos ( \((cos 2\pi + cos \pi) = \) \(\frac 12 \) (1 -1) = 0 (g) cos \(\frac {3\pi}{2} + cos \frac {\pi}{2} \) = 2 cos \(\frac {\frac {3\pi}{2}+\frac {\pi}{2}}{2} cos \frac {\frac {3\pi}{2}-\frac {\pi}{2}}{2} \) = 2 cos π cos π/2 = 2 x -1 x 0 = 0 |
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