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Mathematics

If sec θ - tan θsec θ + tan θ=14,\dfrac{\text{sec θ - tan θ}}{\text{sec θ + tan θ}} = \dfrac{1}{4}, find sin θ.

Trigonometrical Ratios

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Answer

Given, sec θ - tan θsec θ + tan θ=14.\dfrac{\text{sec θ - tan θ}}{\text{sec θ + tan θ}} = \dfrac{1}{4}.

Solving above equation we get,

⇒sec θ - tan θsec θ + tan θ=14⇒1cos θ−sin θcos θ1cos θ+sin θcos θ⇒1 - sin θcos θ1 + sin θcos θ⇒1 - sin θ1 + sin θ=14⇒4(1−sin θ)=1+sin θ⇒4−4 sin θ=1+sin θ⇒sin θ + 4 sin θ=4−1⇒5 sin θ=3⇒sin θ=35.\Rightarrow \dfrac{\text{sec θ - tan θ}}{\text{sec θ + tan θ}} = \dfrac{1}{4} \\[1em] \Rightarrow \dfrac{\dfrac{1}{\text{cos θ}} - \dfrac{\text{sin θ}}{\text{cos θ}}}{\dfrac{1}{\text{cos θ}} + \dfrac{\text{sin θ}}{\text{cos θ}}} \\[1em] \Rightarrow \dfrac{\dfrac{\text{1 - sin θ}}{\text{cos θ}}}{\dfrac{\text{1 + sin θ}}{{\text{cos θ}}}} \\[1em] \Rightarrow \dfrac{\text{1 - sin θ}}{\text{1 + sin θ}} = \dfrac{1}{4} \\[1em] \Rightarrow 4(1 - \text{sin θ}) = 1 + \text{sin θ} \\[1em] \Rightarrow 4 - 4\text{ sin θ} = 1 + \text{sin θ} \\[1em] \Rightarrow \text{sin θ + 4 sin θ} = 4 - 1 \\[1em] \Rightarrow 5 \text{ sin θ} = 3 \\[1em] \Rightarrow \text{sin θ} = \dfrac{3}{5}.

Hence, sin θ = 35\dfrac{3}{5}.

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