Question

How do you evaluate $\sin \left( {\dfrac{{7\pi }}{2}} \right)$ ?

Answer 1

Hint: Here, we will first rewrite the angle as the sum of two angles. Then we will simplify it using the quadrant in which the angle is present. Then we will substitute the value of the obtained angle to get the required value.

Formula Used:
Trigonometric Ratios: $\sin \left( {3\pi + \theta } \right) = - \sin \theta $

Complete step by step solution:
We are given a Trigonometric Ratio$\sin \left( {\dfrac{{7\pi }}{2}} \right)$.
Now, we will simplify the angle, so we get
$ \Rightarrow \sin \left( {\dfrac{{7\pi }}{2}} \right) = \sin \left( {\dfrac{{6\pi + \pi }}{2}} \right)$
Now, by rewriting the trigonometric ratio, we get
$ \Rightarrow \sin \left( {\dfrac{{7\pi }}{2}} \right) = \sin \left( {\dfrac{{6\pi }}{2} + \dfrac{\pi }{2}} \right)$
$ \Rightarrow \sin \left( {\dfrac{{7\pi }}{2}} \right) = \sin \left( {3\pi + \dfrac{\pi }{2}} \right)$
We know that $3\pi + \dfrac{\pi }{2}$ lies in III rd Quadrant. Thus, the sign of Sine in the III rd Quadrant is negative.
We know that Trigonometric Ratios: $\sin \left( {3\pi + \theta } \right) = - \sin \theta $ . Therefore,
$ \Rightarrow \sin \left( {3\pi + \dfrac{\pi }{2}} \right) = - \sin \left( {\dfrac{\pi }{2}} \right)$
We know that the Trigonometric Value of $\sin \left( {\dfrac{\pi }{2}} \right) = 1$. Substituting this value in the above equation, we get
$ \Rightarrow \sin \left( {3\pi + \dfrac{\pi }{2}} \right) = - \left( 1 \right)$
$ \Rightarrow \sin \left( {3\pi + \dfrac{\pi }{2}} \right) = - 1$

Therefore, the value of $\sin \left( {\dfrac{{7\pi }}{2}} \right)$ is$ - 1$.

Additional Information:
We know that Trigonometric Equation is defined as an equation involving trigonometric ratios. Trigonometric identity is an equation that is always true for all the variables. We should know that we have many trigonometric identities that are related to all the other trigonometric equations. Trigonometric Ratios of a Particular angle are the ratios of the sides of a right-angled triangle with respect to any of its acute angle. They are used to find the relationships between the sides of a right-angle triangle.

Note:
We know that all the Trigonometric Ratios are positive in the First Quadrant. Sine and Cosecant are positive in the second quadrant and the rest are negative. Tangent and Cotangent are positive in the third quadrant and the rest are negative. Cosine and Secant are positive in the fourth quadrant and the rest are negative. This can be remembered as the ASTC rule in Trigonometry. This rule is used in determining the signs of the trigonometric ratio.