Question

How do you evaluate \[\sin 5\pi \]?

Answer 1

Hint: We need to find the method to find the value of \[\sin 5\pi \]. Trigonometry is a part of calculus and the basic ratios of trigonometric are sine and cosine which have their application in sound and lightwave theories. The trigonometric have vast applications in naval engineering such as determining the height of the wave and the tide in the ocean.

Complete step by step solution:
In this question, we have given a trigonometric term \[\sin 5\pi \] and we need to obtain the value of the trigonometric term at $5\pi $.
To obtain the value of the trigonometric term we will write the angle as a sum or difference.
Here, \[5\pi \] is written as \[4\pi + \pi \].
Now we will substitute \[4\pi + \pi \] in place of \[5\pi \] in the trigonometric term \[\sin 5\pi \].
This gives,
\[ \Rightarrow \sin \left( {4\pi + \pi } \right)\]
Here, we will calculate the value of \[\sin \pi \] is written as,
\[\sin \pi = \sin \left( {\dfrac{\pi }{2} + \dfrac{\pi }{2}} \right)\]
Since, the value of \[\sin \left( {90 + x} \right) = \cos x\].
Then, \[\sin \left( {\dfrac{\pi }{2} + \dfrac{\pi }{2}} \right) = \cos \dfrac{\pi }{2}\]
The value of \[\cos \dfrac{\pi }{2} = 0\]
Thus, the value of \[\sin \pi = 0\].
The value of \[\sin \pi \]is zero, as the angle pi is revolved four more times the angle is \[5\pi \] which again sums to \[0\].

Thus, the value of \[\sin 5\pi = 0\].

Note:
As we know that the sine angle formula is used to determine the ratio of perpendicular to height in a right-angle triangle. It is also used to determine the missing sides and the angles in other types of triangles.