Question

IF 3 cot theta = 4, find the value of
5 sin theta - 3 cos theta
5 sin theta + 3 cos theta​

Answer 1

Answer

The value of 5sinθ - 3cosθ is 3/5

and 5sinθ + 3cosθ is 27/5

$\bf \large\underline{Given : }$

• 3cosθ = 4

$\bf \large{ \underline{To Find : }}$

• The value of 5sinθ - 3cosθ
• and 5sinθ + 3cosθ

$\bf \large \underline{Solution : }$

Given,

$\sf3\cot\theta = 4 \\\\ \sf\implies \cot\theta = \dfrac{4}{3} \\\\ \sf \underline{we \: know \: that} \\\\ \sf\implies \csc^{2}\theta= \cot^{2}+1\\\\ \sf\implies \csc\theta = \sqrt{ \cot^{2}+1}\\\\ \sf\implies \csc\theta =\sqrt{(\dfrac{4}{3})^{2} + 1}\\\\ \sf\implies \csc\theta = \sqrt{\dfrac{16}{9}+1} \\\\ \sf\implies \csc\theta = \sqrt{\dfrac{25}{9}}\\\\ \sf\implies \csc\theta = \dfrac{5}{3} \\\\ \sf\implies \dfrac{1}{\sin\theta}=\dfrac{5}{3}\\\\ \sf\implies \sin\theta = \dfrac{3}{5}$

Again we have

$\sf 3\cot\theta = 4 \\\\ \sf\implies \cot\theta= \dfrac{4}{3} \\\\ \sf\implies \dfrac{\cos\theta}{\sin\theta}=\dfrac{4}{3} \\\\ \implies 3\cos\theta= 4\sin\theta$

Therefore ,

$\sf\longrightarrow 5\sin\theta - 3\cos\theta \\\\ \sf\longrightarrow 5\sin\theta - 4\cos\theta \: \: \{ \because 3\cos\theta= 4\sin\theta \} \\\\ \sf\longrightarrow \sin\theta \\\\ \sf\longrightarrow \dfrac{3}{5}$

And

$\sf\longrightarrow 5\sin\theta + 3\cos\theta \\\\ \sf\longrightarrow 5\sin\theta+4\sin\theta \\\\ \sf\longrightarrow 9\sin\theta \\\\ \sf\longrightarrow 9\times \dfrac{3}{5} \\\\ \sf\longrightarrow \dfrac{27}{5}$

Answer 2

Explanation:

$\bf \large\underline{Solution : }$