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Geometry Level 3

If 3 cos ⁡ θ − 5 sin ⁡ θ = 3 2 3\cos\theta-5\sin\theta = 3\sqrt{2} , where θ ∈ ( − π 2 , 0 ) \theta \in ( - \frac{ \pi}{2} , 0 ) , then what is the value of 5 cos ⁡ θ + 3 sin ⁡ θ − 4 5\cos\theta+3\sin\theta-4 ?


The answer is 0.

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Gagan Raj by Gagan Raj , 23, India

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1 solution

Gagan Raj
Apr 7, 2015

3 cos ⁡ θ − 5 sin ⁡ θ = 3 2 3\cos\theta-5\sin\theta=3\sqrt{2} for the given condition on θ \theta .

Now , ( 3 cos ⁡ θ − 5 sin ⁡ θ ) 2 (3\cos\theta-5\sin\theta)^{2} + + ( 5 cos ⁡ θ + 3 sin ⁡ θ ) 2 (5\cos\theta+3\sin\theta)^{2} = 9 + 25 =9+25

18 + ( 5 cos ⁡ θ + 3 sin ⁡ θ ) 2 18+(5\cos\theta+3\sin\theta)^{2} = 34 =34

5 cos ⁡ θ + 3 sin ⁡ θ 5\cos\theta+3\sin\theta = = 34 − 18 \sqrt{34-18} = = 16 \sqrt{16} = 4 =4

5 cos ⁡ θ + 3 sin ⁡ θ − 4 = 4 − 4 = 0 5\cos\theta+3\sin\theta-4=4-4=0

2 Helpful 0 Interesting 0 Brilliant 0 Confused

If 0 < θ < π / 2 0 < \theta < \pi/2 , then cos ⁡ θ < 1 \cos \theta < 1 and sin ⁡ θ > 0 \sin \theta > 0 , so 3 cos ⁡ θ − 5 sin ⁡ θ < 3 , 3 \cos \theta - 5 \sin \theta < 3, and it cannot be equal to 3 2 3 \sqrt{2} .

Jon Haussmann - 6 years, 2 months ago

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Thanks. I have updated the condition accordingly.

@Gagan Raj Can you udpate your solution too?

Calvin Lin Staff - 6 years, 2 months ago

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