Question

How do you evaluate \[\sin (\dfrac{\pi }{2})\]?

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 \[0{\text{ }}to{\text{ }}90\], 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 \[\sin (\dfrac{\pi }{2})\]
If in a right angled triangle θ represents one of its acute angle then by definition we can write
\[\sin \theta = \dfrac{{Opposite}}{{Hypotenuse}}\]
Now if we increase the magnitude of \[\theta \] slowly, then the magnitude of the side opposite to it will slowly increase and thus, the magnitude of adjacent will go to decrease. When \[\theta \] becomes equal to \[\dfrac{\pi }{2}\] , then the adjacent side will vanish and the opposite side will coincide with the hypotenuse.
So, we come to a conclusion that when \[\theta = \dfrac{\pi }{2}\], then “side opposite to \[\theta \]” \[ = \] ”hypotenuse”.
Therefore,
\[\sin \theta = \dfrac{{Opposite}}{{Hypotenuse}}\]
\[ \Rightarrow \sin (\dfrac{\pi }{2}) = \dfrac{{Hypotenuse}}{{Hypotenuse}} = 1\].

Hence, the exact value of \[\sin (\dfrac{\pi }{2})\] is \[1\].

Note:
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 θ represents one of its acute angles then the sine of theta is the ratio of the opposite side to the hypotenuse.