Math, asked by raziairsath27, 1 year ago

Cosecant theta = √10 find secant theta​

Answers

Answered by Anonymous
5

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.

Answered by Anonymous
6

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} }

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