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If 5 Secθ – 12 Cosecθ = 0, Find the Values of Secθ, Cosθ and Sinθ. - Geometry

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Question

If 5 secθ – 12 cosecθ = 0, find the values of secθ, cosθ and sinθ.

Solution

\[5\sec\theta - 12cosec\theta = 0\]

\[ \Rightarrow 5\sec\theta = 12cosec\theta\]

\[ \Rightarrow \frac{5}{\cos\theta} = \frac{12}{\sin\theta}\]

\[ \Rightarrow \frac{\sin\theta}{\cos\theta} = \frac{12}{5}\]

\[ \Rightarrow \tan\theta = \frac{12}{5}\]
We have,
\[\sec^2 \theta = 1 + \tan^2 \theta\]
\[ \Rightarrow \sec^2 \theta = 1 + \left( \frac{12}{5} \right)^2 \]
\[ \Rightarrow \sec^2 \theta = 1 + \frac{144}{25} = \frac{169}{25}\]
\[ \Rightarrow \sec\theta = \sqrt{\frac{169}{25}} = \frac{13}{5}\]
Now,
\[\sec^2 \theta = 1 + \tan^2 \theta\]
\[ \Rightarrow \sec^2 \theta = 1 + \left( \frac{12}{5} \right)^2 \]
\[ \Rightarrow \sec^2 \theta = 1 + \frac{144}{25} = \frac{169}{25}\]
\[ \Rightarrow \sec\theta = \sqrt{\frac{169}{25}} = \frac{13}{5}\]
Also,
\[\frac{\sin\theta}{\cos\theta} = \tan\theta\]
\[ \Rightarrow \sin\theta = \tan\theta \times \cos\theta\]
\[ \Rightarrow \sin\theta = \frac{12}{5} \times \frac{5}{13} = \frac{12}{13}\]
Thus, the values of secθ, cosθ and sinθ are \[\frac{13}{5}\],

\[\frac{5}{13}\] and \[\frac{12}{13}\], respectively.
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Solution If 5 Secθ – 12 Cosecθ = 0, Find the Values of Secθ, Cosθ and Sinθ. Concept: Application of Trigonometry.
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