Question

What is the value of $\sin \left( {2A} \right)$ , if A is $30$ degrees ?

Answer 1

Hint : The given problem requires us to find the value of a trigonometric expression. The question requires thorough knowledge of trigonometric functions, formulae and identities. The question describes the wide ranging applications of trigonometric identities and formulae. We must keep in mind the trigonometric identities while solving such questions.

Complete step-by-step answer :
In the given question, we are required to evaluate the value of $\sin \left( {2A} \right)$ when we are provided the value of angle A is $30$ degrees using the basic concepts of trigonometry and identities.
So, we substitute the value of the angle A in the expression whose value has to be evaluated.
So, we get, $\sin \left( {2A} \right)$
$\Rightarrow \sin \left( {2 \times {{30}^ \circ }} \right)$
Simplifying the calculation of the angle I the bracket, we get,
$\Rightarrow \sin \left( {{{60}^ \circ }} \right)$
Now, we have to find the value of the trigonometric function sine of angle ${60^ \circ }$ .
We know that the value of $\sin \left( {{{60}^ \circ }} \right)$ is $\left( {\dfrac{{\sqrt 3 }}{2}} \right)$ .
So, we get,
$\Rightarrow \sin \left( {{{60}^ \circ }} \right) = \left( {\dfrac{{\sqrt 3 }}{2}} \right)$
Hence, we get the value of $\sin \left( {2A} \right)$ , if A is $30$ degrees as $\left( {\dfrac{{\sqrt 3 }}{2}} \right)$ .
So, the correct answer is $\left( {\dfrac{{\sqrt 3 }}{2}} \right)$ ”.

Note : There are six trigonometric ratios: $\sin \theta$ , $\cos \theta$ , $\tan \theta$ , $\cos ec\theta$ , $\sec \theta$ and $\cot \theta$ . Basic trigonometric identities include ${\sin ^2}\theta + {\cos ^2}\theta = 1$ , ${\sec ^2}\theta = {\tan ^2}\theta + 1$ and $\cos e{c^2}\theta = {\cot ^2}\theta + 1$ . These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above. The given problem involves the use of trigonometric formulae and identities. Such questions require thorough knowledge of trigonometric conversions and ratios. Algebraic operations and rules like transposition rule come into significant use while solving such problems.