Question

Find the value of ${\theta }_1+{\theta }_2+{\theta }_3$ in each of the following

(1) 2 sin 2${\theta }_1$ = $\sqrt{3}$

(2) 2 cos 3${\theta }_2$ = 1

(3) $\sqrt{3}$tan 2${\theta }_3$ - 3 = 0

• A. 60°​
• B. 80°​
• C. 90°​
• D. None of the given option

Answer 1

The correct option is B

80°​

(1) we have,
2sin 2${\theta }_1$ = $\sqrt{3}$

$\Rightarrow$ sin 2${\theta }_1$ = $\frac{\sqrt{3}}{2}$

$\Rightarrow$ sin 2${\theta }_1$ = sin 2${60}^{\circ }$

$\Rightarrow$ ${\theta }_1$ = ${30}^{\circ }$

(2) we have, 2cos 3${\theta }_2$ = 1

$\Rightarrow$ cos3${\theta }_2$ = $\frac{1}{2}$

$\Rightarrow$ coss 3${\theta }_2$ = cos ${60}^{\circ }$

$\Rightarrow$ 3${\theta }_2$ = ${60}^{\circ }$

$\Rightarrow$ ${\theta }_2$= ${20}^{\circ }$

(3) we have, $\sqrt{3}$tan 2${\theta }_3$ - 3 = 0

$\Rightarrow$ $\sqrt{3}$tan 2${\theta }_3$ = 3

$\Rightarrow$ tan 2${\theta }_3$ = $\frac{3}{\sqrt{3}}$ = $\sqrt{3}$

$\Rightarrow$ tan 2${\theta }_3$ = tan ${60}^{\circ }$

$\Rightarrow$ 2${\theta }_3$ = ${60}^{\circ }$

$\Rightarrow$ $\theta$ = ${30}^{\circ }$

Hence, value of the sum ​${\theta }_1$ + ${\theta }_2$ + ${\theta }_3$ = $({30}^{\circ }+{20}^{\circ }+{30}^{\circ })={80}^{\circ }$