Let's try to solve this Equation

Geometry Level 3

If cos ⁡ ( 2 x ) = − 2 3 \cos(2x)=-\dfrac23 and P = 2 − 3 cos ⁡ x cos ⁡ x 3 + 4 sin ⁡ x sin ⁡ x P=\dfrac{2-3\cos x \cos x} {3+4\sin x \sin x} .

Find P P (to four decimal places).


The answer is 0.2368.

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Cong Thanh by Cong Thanh , 22, Vietnam

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2 solutions

Chew-Seong Cheong
Jun 10, 2016

P = 1 − 3 cos ⁡ x cos ⁡ x 3 + 4 sin ⁡ x sin ⁡ x = 1 − 3 cos ⁡ 2 x 3 + 4 sin ⁡ 2 x = 1 − 3 2 ( 1 + cos ⁡ 2 x ) 3 + 4 2 ( 1 − cos ⁡ 2 x ) = 1 − 3 2 ( 1 − 2 3 ) 3 + 4 2 ( 1 + 2 3 ) = 1 − 3 2 ( 1 3 ) 3 + 4 2 ( 5 3 ) = 1 − 3 2 ( 1 3 ) 3 + 4 2 ( 5 3 ) = 1 2 19 3 = 3 38 ≈ 0.2368 \begin{aligned} P & = \frac{1-3\cos x \cos x}{3+4\sin x \sin x} = \frac{1-3\cos^2 x}{3+4\sin^2 x} = \frac{1-\frac32 \left(1+\cos 2 x\right)}{3+\frac42 \left(1-\cos 2 x\right)} \\ & = \frac{1-\frac32 \left(1-\frac23 \right)}{3+\frac42 \left(1+\frac23 \right)} = \frac{1-\frac32 \left(\frac13 \right)}{3+\frac42 \left(\frac53 \right)} = \frac{1-\frac32 \left(\frac13 \right)}{3+\frac42 \left(\frac53 \right)} \\ & = \frac{\frac12}{\frac{19}3} = \frac3{38} \approx \boxed{0.2368} \end{aligned}

2 Helpful 0 Interesting 1 Brilliant 0 Confused
Ahmed Hossain
Jun 13, 2016

ans is 9 38 \frac{9}{38}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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