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19/01/2022 4:25 pm
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Let S be the sum of all solutions (in radians) of the equation sin4θ + cos4θ – sinθ cosθ = 0 in [0, 4π]. Then 8S/π is equal to ..........
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19/01/2022 4:34 pm
We know that,
sin4θ + cos4θ – sinθ cosθ = 0
⇒ 1 - sin2θ cos2θ - sinθ cosθ = 0
⇒ 2 -(sin 2θ)2 - sin 2θ = 0
⇒ (sin 2θ)2 - (sin 2θ) - 2 = 0
⇒ (sin 2θ + 2) - (sin 2θ - 1) = 0
⇒ sin 2θ = 1 or sin 2θ = -2
⇒ 2θ = \(\frac{\pi}{2}, \frac{5\pi}{2}, \frac{9 \pi}{2}, \frac{13 \pi}{2}\)
⇒ θ = \(\frac{\pi}{4}, \frac{5\pi}{4}, \frac{9 \pi}{4}, \frac{13 \pi}{4}\)
⇒ S = \(\frac{\pi}{4}+ \frac{5\pi}{4}+ \frac{9 \pi}{4}+ \frac{13 \pi}{4}\) = 7π
⇒ \(\frac{8S}{\pi}\) = \(\frac{8 \times 7 \pi}{\pi}\) = 56.00
This post was modified 3 years ago by admin