Question

Verify the following
$2\sin {{30}^{0}}\cos {{30}^{0}}=\sin {{60}^{0}}$

Answer 1

Hint: The Trigonometric ratios table helps to find the values of trigonometric standard angles such as ${{0}^{0}},{{30}^{0}},{{45}^{0}},{{60}^{0}}$ and ${{90}^{0}}$. It consists of trigonometric ratios – sine, cosine, tangent, cosecant, secant and cotangent. These ratios can be written in short as sin, cos, tan, cosec, sec and cot.

Complete step-by-step answer:
The value of the trigonometric ratios by using the trigonometric table is given below.

$\sin {{60}^{0}}=\dfrac{\sqrt{3}}{2},\sin {{30}^{0}}=\dfrac{1}{2},\cos {{30}^{0}}=\dfrac{\sqrt{3}}{2}$

Let us consider the left side of the given expression

\[2\sin {{30}^{0}}\cos {{30}^{0}}=2\times \dfrac{1}{2}\times \dfrac{\sqrt{3}}{2}\]

Cancelling the terms 2 on the right side, we get

\[2\sin {{30}^{0}}\cos {{30}^{0}}=\dfrac{\sqrt{3}}{2}\]

From the trigonometric table, $\dfrac{\sqrt{3}}{2}=\sin {{60}^{0}}$

$2\sin {{30}^{0}}\cos {{30}^{0}}=\sin {{60}^{0}}$

Hence the given expression is verified


Note: Alternatively, the given question is verified by using the sine double angle formula that tells us that $\sin 2\theta $ is equal to $2\sin \theta \cos \theta $ that is $\sin 2A=2\sin A\cos A$. Let $A={{30}^{0}}$, then. $2\sin {{30}^{0}}\cos {{30}^{0}}=\sin \left( 2\times 30 \right)=\sin {{60}^{0}}$.