Question

Evaluate : sin 60° + cos 30° + sin 30°cos30°

Answer 1

Answer :-

sin60° + cos30° + sin30° cos30° = 5√3/4.

Step-by-step explanation:

To Find :-

• Evaluate: sin60° + cos30° + sin30° cos30°

★ Solution

$\sf{\longrightarrow \: {sin60}^{\circ} + {cos30}^{\circ} + {sin30}^{\circ} {cos30}^{\circ}}$

First solving, [sin30° cos30°],

By using the trigonometric ratio, [sin30° = 1/2] and [cos30° = √3/2],

$\sf{\longrightarrow \: {sin60}^{\circ} + {cos30}^{\circ} + \dfrac{1}{2} \times \dfrac{ \sqrt{3}}{2}}$

$\sf{\longrightarrow \: {sin60}^{\circ} + {cos30}^{\circ} + \dfrac{1 \times \sqrt{3}}{2 \times 2}}$

$\sf{\longrightarrow \: {sin60}^{\circ} + {cos30}^{\circ} + \dfrac{ \sqrt{3}}{4}}$

Now, solving [sin60° + cos30°],

By using the trigonometric ratio, [sin60° = √3/2] and [cos30° = 1/2],

$\sf{\longrightarrow \: \dfrac{ \sqrt{3}}{2} + \dfrac{ \sqrt{3}}{2} + \dfrac{ \sqrt{3}}{4}}$

$\sf{\longrightarrow \: \dfrac{ \sqrt{3} + \sqrt{3}}{2} + \dfrac{ \sqrt{3}}{4}}$

$\sf{\longrightarrow \: \dfrac{2 \sqrt{3}}{2} + \dfrac{ \sqrt{3}}{4}}$

$\sf{\longrightarrow \: \dfrac{ \not{2} \sqrt{3}}{ \not{2}} + \dfrac{ \sqrt{3}}{4}}$

$\sf{\longrightarrow \: \sqrt{3} + \dfrac{ \sqrt{3}}{4}}$

By simplifying, [√3 + √3/4],

$\sf{\longrightarrow \: \sqrt{3} + \dfrac{ \sqrt{3}}{4}}$

$\sf{\longrightarrow \: \dfrac{ \sqrt{3}}{1} + \dfrac{ \sqrt{3}}{4}}$

$\sf{\longrightarrow \: \dfrac{4\sqrt{3} + \sqrt{3}}{4}}$

$\sf{\longrightarrow \: \dfrac{5 \sqrt{3}}{4}}$

$\bf{\longrightarrow \: \dfrac{5 \sqrt{3}}{4}}$

Thus, sin60° + cos30° + sin30° cos30° = 5√3/4.