Question

cot theta = 2tan 7and 1/2 ÷1-tan square 7 and 1/2 then sin3theta=​

Answer 1

please refer the attachment

Answer 2

Concept:

We will use the following formulas to solve this question.

•           cot $\theta$ = tan (90 - $\theta$ )            (1)

•        tan 2 x = $\frac {2 tan x}{1- tan^{2} x}$                   (2)

• sin (180 + x) = -sin x                      (3)

Given :

The trigonometric relation cot $\theta$ = $\frac {2 tan 7\frac{1}{2} ^0}{1- tan^{2} 7\frac{1}{2} ^0}$ .

To find:

The value of the trigonometric function sin 3 $\theta$.

Solution:

The given relation can be written as:

cot $\theta$ = $\frac {2 tan \frac{15}{2} ^0}{1- tan^{2} \frac{15}{2} ^0}$

On comparing with the formula of tan 2x given by equation (2) we get

cot $\theta$ = tan 2 × $\frac{15}{2}^0$

cot $\theta$ = tan $15^{0}$

cot $\theta$ = tan $(90-75)^{0}$

Now using formula of equation (2)

cot $\theta$  = cot $75^{0}$

So,   $\theta$ = $75^{0}$.

then, sin 3 $\theta$ = sin(3 × $75^{0}$)

                    = sin $225^{0}$

                    = sin $(180 ^0+45^0)$

Using formula of equation (3)

                    = - sin $45^0$

                    =  $-\frac{1}{\sqrt{2} }$.

Hence, the value of sin 3 $\theta$ = $-\frac{1}{\sqrt{2} }$.

Option(1) is the correct choice.