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

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ConceptApplication of Trigonometry

#### 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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#### APPEARS IN

Solution If 5 Secθ – 12 Cosecθ = 0, Find the Values of Secθ, Cosθ and Sinθ. Concept: Application of Trigonometry.
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