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Geometry Level 3

If 3 sin ⁡ θ + 5 cos ⁡ θ = 5 3\sin \theta + 5\cos \theta = 5 , where θ ≠ 0 \theta \ne 0 , compute 5 sin ⁡ θ − 3 cos ⁡ θ 5\sin\theta - 3\cos\theta .


The answer is 3.

Mohan Aditya by Mohan Aditya , 21, India

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1 solution

Reineir Duran
Mar 27, 2016

Let x = 5 sin ⁡ θ − 3 cos ⁡ θ x = 5\sin \theta - 3\cos \theta . Notice that 5 2 + x 2 = 9 sin ⁡ 2 θ + 30 sin ⁡ θ cos ⁡ θ + 25 cos ⁡ 2 θ + 25 sin ⁡ 2 θ − 30 sin ⁡ θ cos ⁡ θ + 9 cos ⁡ 2 θ = 34 ( sin ⁡ 2 θ + cos ⁡ 2 θ ) = 34. \begin{aligned} 5^2 + x^2 &= 9\sin^2 \theta + 30\sin \theta\cos \theta + 25\cos^2 \theta + 25\sin^2 \theta - 30\sin \theta \cos \theta + 9\cos^2 \theta \\ &= 34\left(\sin^2 \theta + \cos^2 \theta\right) \\ &= 34. \end{aligned}

From here, we get x = ± 3 x = \pm 3 . Since θ ≠ 0 \theta \neq 0 , 5 sin ⁡ θ − 3 cos ⁡ θ 5\sin \theta - 3\cos \theta cannot be − 3 -3 , and thus x = 3 \boxed{x = 3} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Mention why x ≠ − 3 \color{magenta}{x ≠ -3} .

Aditya Sky - 5 years, 2 months ago

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Since θ ≠ 0 \theta \neq 0 , 5 sin ⁡ θ − 3 cos ⁡ θ 5\sin \theta - 3\cos \theta cannot be − 3 -3 .

Reineir Duran - 5 years, 2 months ago

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Exactly !. You should mention this in your solution.

Aditya Sky - 5 years, 2 months ago

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@Aditya Sky – Oh sorry... I thought this will be trivial for everyone.

Reineir Duran - 5 years, 2 months ago

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