Question

Cosecant theta = √10 find secant theta​

Answer 1

Solution :-

cosecθ = √10

We know that

$\rm cosec^{2} \theta - cot^{2} \theta = 1$

$\implies \rm cosec^{2} \theta - \bigg( \dfrac{1}{tan \theta} \bigg)^{2} = 1$

[ Because cotθ = 1/tanθ ]

$\implies \rm cosec^{2} \theta - \dfrac{1}{tan^{2} \theta} = 1$

Substituting cosecθ = √10

$\implies \rm ( \sqrt{10}) ^{2} - \dfrac{1}{tan^{2} \theta} = 1$

$\implies \rm 10- \dfrac{1}{tan^{2} \theta} = 1$

$\implies \rm 10-1 = \dfrac{1}{tan^{2} \theta}$

$\implies \rm 9= \dfrac{1}{tan^{2} \theta}$

$\implies \rm tan^{2} \theta = \dfrac{1}{9}$

We know that

$\rm sec^{2} \theta - tan^{2} \theta = 1$

Substituting tan²θ = 1/9

$\implies \rm sec^{2} \theta - \dfrac{1}{9} = 1$

$\implies \rm sec^{2} \theta = 1 + \dfrac{1}{9}$

$\implies \rm sec^{2} \theta = \dfrac{10}{9}$

Taking square root on both sides

$\implies \rm sec\theta = \dfrac{ \sqrt{10}}{3}$

Therefore the value of secθ is √10/3.

Answer 2

Answer:

$\large\boxed{\sf{\dfrac{\sqrt{10}}{3}}}$

Step-by-step explanation:

Given,

$\csc( \theta) = \sqrt{10}$

To find

$\sec( \theta )$

We know that,

$\sin( \theta ) = \frac{1}{ \csc( \theta ) } \\ \\ = > \sin( \theta ) = \frac{1}{ \sqrt{10} }$

Also, we know that,

$\cos( \theta ) = \sqrt{1 - { \sin }^{2} \theta } \\ \\ = > \cos( \theta ) = \sqrt{1 - {( \frac{1}{ \sqrt{10} } )}^{2} } \\ \\ = > \cos( \theta ) = \sqrt{1 - \frac{1}{10} } \\ \\ = > \cos( \theta ) = \sqrt{ \frac{9}{10} } \\ \\ = > \cos( \theta ) = \frac{ 3}{ \sqrt{10} }$

But, we know that,

$\sec( \theta ) = \frac{1}{ \cos( \theta ) } \\ \\ = > \sf{\sec( \theta ) = \frac{ \sqrt{10} }{3} }$