Question

if cos theta is equals to root 3 by 2 then find 4 sin square theta + tan square theta​

Answer 1

Given:

$\cos \: theta \: = \frac{ \sqrt{3} }{2}$

To find:

$4 \: {sin}^{2} \: theta + {tan}^{2} \: theta$

Solution:

$\cos \: theta \: = \frac{ \sqrt{3} }{2}$

We know that:

${sin}^{2} \: theta = 1 - {cos}^{2} \: theta$

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

$= > {sin}^{2} \: theta = 1 - \frac{3}{4}$

$= > {sin}^{2} \: theta = \frac{1}{4}$

$= > {sin} \: theta = \sqrt{ \frac{1}{4} }$

=> sin theta = 1/2

We know,

tan theta = sin theta / cos theta

= (1/2) / (✓3/2)

= 1/✓3

${tan}^{2} \: theta = \frac{1}{3}$

Therefore,

$4 \: {sin}^{2} \: theta + {tan}^{2} \: theta$

$= 4 \times \frac{1}{4} + \frac{1}{3}$

$= 1 + \frac{1}{3}$

$= \frac{3 + 1}{3}$

$= \frac{4}{3}$

Therefore, the value of the given expression is 4/3.