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Question: If \(5\tan\theta = 4,\) then \(\frac{5\sin\theta - 3\cos\theta}{5\sin\theta + 2\cos\theta} =\)...

If 5tan⁡θ=4,5\tan\theta = 4, then 5sin⁡θ−3cos⁡θ5sin⁡θ+2cos⁡θ=\frac{5\sin\theta - 3\cos\theta}{5\sin\theta + 2\cos\theta} =

A

0

B

1

C

1/6

D

6

Answer

1/6

Explanation

Solution

5tan⁡θ=4⇒tan⁡θ=455\tan\theta = 4 \Rightarrow \tan\theta = \frac{4}{5}

∴sin⁡θ=441\therefore\sin\theta = \frac{4}{\sqrt{41}}and cos⁡θ=541\cos\theta = \frac{5}{\sqrt{41}}

5sin⁡θ−3cos⁡θ5sin⁡θ+2cos⁡θ=5×441−3×5415×441+2×541\frac{5\sin\theta - 3\cos\theta}{5\sin\theta + 2\cos\theta} = \frac{5 \times \frac{4}{\sqrt{41}} - 3 \times \frac{5}{\sqrt{41}}}{5 \times \frac{4}{\sqrt{41}} + 2 \times \frac{5}{\sqrt{41}}} 20−1520+10=530=16\frac{20 - 15}{20 + 10} = \frac{5}{30} = \frac{1}{6}.