Trigonometry!!

Geometry Level 2

Suppose a a is a real number such that:

sin ⁡ ( π cos ⁡ a ) = cos ⁡ ( π sin ⁡ a ) \sin(\pi \cos a) = \cos(\pi \sin a) .

Evaluate 35 sin ⁡ 2 ( 2 a ) + 84 cos ⁡ 2 ( 4 a ) = ? 35 \sin^2(2a) + 84 \cos^2(4a)=?

20 24 30 34 21

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

From the given equation we get sin ⁡ a + cos ⁡ a = 1 2 \sin a+\cos a=\dfrac{1}{2} , or sin ⁡ 2 a = − 3 4 \sin {2a}=-\dfrac{3}{4} . So cos ⁡ 4 a = − 1 8 \cos {4a}=-\dfrac{1}{8} . Therefore the given expression equals 35 × 9 16 + 84 × 1 64 = 21 35\times \dfrac{9}{16}+84\times \dfrac{1}{64}=\boxed {21}

2 Helpful 0 Interesting 1 Brilliant 0 Confused
Amal Hari
Nov 9, 2019

cos ⁡ ( π 2 − θ ) = sin ⁡ θ \cos \left( \frac{\pi}{2} -\theta\right) =\sin \theta

cos ⁡ ( π 2 − π cos ⁡ a ) = cos ⁡ ( π sin ⁡ a ) \cos\left(\frac{\pi}{2} -\pi \cos a\right) =\cos\left( \pi \sin a\right)

( π 2 − π cos ⁡ a ) = ( π sin ⁡ a ) \left(\frac{\pi}{2} -\pi \cos a\right) =\left( \pi \sin a\right)

π 2 = π sin ⁡ a + π cos ⁡ a \frac{\pi}{2} =\pi \sin a+\pi \cos a

remove π \pi and square equation:

1 4 = 1 + sin ⁡ 2 a \frac{1}{4} =1+ \sin 2a

sin ⁡ 2 a = − 3 4 \sin 2a = -\frac{3}{4}

solving for a we will get approximately − 24.29 5 ∘ -24.295 ^{\circ}

evaluating the expression with this value gives 20.9999 ≈ 21 20.9999 \approx 21

2 Helpful 0 Interesting 1 Brilliant 0 Confused

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