maxiii

Geometry Level 3

Whats the maximum value of

( 5 sin ⁡ x − 12 cos ⁡ x ) ( 5 cos ⁡ x + 12 sin ⁡ x ) (5\sin x -12\cos x )(5\cos x +12\sin x)


The answer is 84.5.

Mayank Garg by mayank garg , 21, India

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2 solutions

Pranjal Jain
Dec 25, 2014

Let cos ⁡ θ = 5 13 \cos \theta=\dfrac{5}{13}

( 5 sin ⁡ x − 12 cos ⁡ x ) ( 5 cos ⁡ x + 12 sin ⁡ x ) = 169 ( cos ⁡ θ sin ⁡ x − sin ⁡ θ cos ⁡ x ) ( cos ⁡ θ cos ⁡ x + sin ⁡ θ sin ⁡ x ) = 169 sin ⁡ ( x − θ ) c o s ( x − θ ) = 84.5 ( sin ⁡ ( 2 x − 2 θ ) ) (5\sin x-12\cos x)(5\cos x+12\sin x)\\=169(\cos\theta\sin x-\sin\theta\cos x)(\cos\theta\cos x+\sin\theta\sin x)\\=169\sin(x-\theta)cos(x-\theta)\\=84.5(\sin(2x-2\theta))

So maximum value is 84.5

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K V Shenoy
Dec 25, 2014

( 5 sin ⁡ x − 12 cos ⁡ x ) ( 5 cos ⁡ x + 12 sin ⁡ x ) (5\sin x - 12\cos x)(5\cos x + 12\sin x) = { 60 ( sin ⁡ 2 x − cos ⁡ 2 x ) − 119 sin ⁡ x cos ⁡ x } \{60(\sin^{2}x-\cos^{2}x)-119 \sin x \cos x\} = − 60 cos ⁡ 2 x − 59.5 sin ⁡ 2 x -60\cos 2x -59.5\sin 2x .

The maximum value of this expression is 6 0 2 + 59. 5 2 = 84.5 □ \sqrt{60^{2} + 59.5^{2}} = \boxed{84.5} _\square

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