Question

How do you evaluate $\sin (\dfrac{\pi }{3})?$

Answer 1

Hint: Real functions which relate any angle of a right angled triangle to the ratio of any two of its sides are Trigonometric functions. We can also use geometric definitions to evaluate trigonometric values. We need to note that the given value is in the range of 0 to 90, so we will directly imply the cosine property to attain the answer. Here, it’s important that we know the sine of theta is the ratio of the opposite side to the hypotenuse.

Complete step by step solution:
According to the given data, we need to evaluate $\sin (\dfrac{\pi }{3})$
If in a right angled triangle θ represents one of its acute angle then by definition we can write
$\sin \theta = \dfrac{{{\text{Opposite}}}}{{{\text{Hypotenuse}}}}$
When θ becomes equal to $\dfrac{\pi }{3}$ ,the height will have length of $\sqrt 3 $ and hypotenuse is become 2

$\sin \left( {\dfrac{\pi }{3}} \right) = \dfrac{{opposite\dfrac{\pi }{3}}}{{{\text{Hypotenuse}}}} = \dfrac{{\sqrt 3 }}{2}$

Note: One should be careful while evaluating trigonometric values and rearranging the terms to convert from one function to the other. Trigonometric functions are real functions which relate any angle of a right angled triangle to the ratio of any two of its sides. The widely used ones are sin, cos and tan. While the rest can be referred to as the reciprocal of the others, i.e., cosec, sec and cot respectively. If in a right angled triangle $\theta$ represents one of its acute angles then the sine of theta is the ratio of the opposite side to the hypotenuse.