Question

If sin theta=p/q find the value of all trignometric ratios

Answer 1

let , p = perpendicular , q = hypotenuse

r = base

if sinA = p / q

then , => cosecA = 1/ sinA = q / p

= > cosA = √ ( 1 - sin^2 A )

=> cosA = √ ( 1 - p^2 / q^2 )

=> cosA = √{ ( q^2 - p^2 ) / q^2 }

=> cosA = √( r^2 / q^2 )

=> cosA = r / q

=> Sec A = 1 / cosA = 1 / r / q

=> Sec A = q / r

=> tan A = sinA / cosA

=> = ( p / q ) / ( r / q )

=> = p / r

so , Tan A = p / r

=> Cot A = 1/ tanA

=> Cot A = 1 / ( p / r )

=> Cot A = r / q

hence , values of all trigonometric ratios are,

sin A = p / q , cos A = r / q , tan A = p / r

cot A = r / p and cosecA = q / p .
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【 hope helps 】
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Answer 2

Answer:

Given Sin( $\theta$ )= $\frac{p}{q}$ .-------------------------------------------------( 1 )

The formula of    $sin(\theta)=\frac{Opposite side}{Hypotenuse side}$  -----------------( 2 )

To find : cosec ( $\theta$ ), cos( $\theta$ ), sec( $\theta$ ), tan( $\theta$ ) and cot( $\theta$ ).

From equations ( 1 ) and ( 2 ),

Opposite side = p,

Hypotenuse side = q.

$cosec( \theta ) =\frac{1}{sin(\theta)} =\frac{q}{p}$.

Now, using the formula  

                           $sin^2 (\theta)+cos^2(\theta)=1,$

                           $cos(\theta)=\sqrt{1-sin^2(\theta)}$,

Substituting ( 1 ), we get

                           $cos(\theta)=\sqrt{1-\frac{p^2}{q^2} }$,

                                      $=\sqrt{\frac{q^2-p^2}{q^2} }$ ,

                                      $=\sqrt{\frac{z^2}{q^2} }$ , where $z^2=q^2-p^2$.

Therefore, $cos(\theta)=\frac{z}{q}$.---------------------------------------------( 3 )

$sec( \theta ) =\frac{1}{cos(\theta)} =\frac{1}{\frac{z}{q}}=\frac{q}{z}$ .

$tan(\theta)=\frac{sin(\theta)}{cos(\theta)}$

Using ( 1 ) and ( 3 ),

$tan(\theta)=\frac{\frac{p}{q}}{\frac{z}{q}}=\frac{p}{z} .$

Therefore, $tan(\theta)=\frac{p}{z} .$

$cot(\theta)=\frac{1}{tan(\theta)}=\frac{1}{\frac{p}{z}}$,

         $=\frac{z}{p}$

Hence, values of all trigonometric ratios are

$cosec( \theta ) =\frac{q}{p}$, $cos(\theta)=\frac{z}{q}$, $sec( \theta ) =\frac{q}{z}$, $tan(\theta)=\frac{p}{z}$, $cot(\theta)=\frac{z}{p}$.