Question

How do you evaluate \[4{\cos ^2}\left( { - \dfrac{\pi }{2}} \right)\] ?

Answer 1

Hint: Here, we will first rewrite the expression and substitute the value of the given angle. Then we will simplify the expression to get the required value. Trigonometric Ratios of a particular angle are the ratios of the sides of a right-angled triangle with respect to any of its acute angles.

Formula Used:
Trigonometric ratio \[\cos \left( {\dfrac{\pi }{2}} \right) = \cos \left( { - \dfrac{\pi }{2}} \right) = 0\]

Complete Step by Step Solution:
We have to evaluate the given trigonometric ratio \[4{\cos ^2}\left( { - \dfrac{\pi }{2}} \right)\].
Rewriting the expression, we get
\[4{\cos ^2}\left( { - \dfrac{\pi }{2}} \right) = 4{\left( {\cos \left( { - \dfrac{\pi }{2}} \right)} \right)^2}\]
We know that Trigonometric ratio \[\cos \left( {\dfrac{\pi }{2}} \right) = \cos \left( { - \dfrac{\pi }{2}} \right) = 0\]
Now substituting this value in the above equation, we get
\[ \Rightarrow 4{\cos ^2}\left( { - \dfrac{\pi }{2}} \right) = 4{\left( 0 \right)^2}\]
Now, by simplifying the equation, we get
\[ \Rightarrow 4{\cos ^2}\left( { - \dfrac{\pi }{2}} \right) = 4\left( 0 \right)\]
\[ \Rightarrow 4{\cos ^2}\left( { - \dfrac{\pi }{2}} \right) = 0\]

Therefore, the value of \[4{\cos ^2}\left( { - \dfrac{\pi }{2}} \right)\] is 0.

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 are used to find the relationships between the sides of a right-angle triangle.

Note:
We should note in particular that sine and tangent are odd functions since both the functions are symmetric about the origin. Cosine is an even function because the functions are symmetric about the\[y\] axis. So, we take the arguments in the negative sign for odd functions and positive signs for even functions. We should remember that any multiple of \[\pi \] and \[ \pm \dfrac{\pi }{2}\] gives a cosine equal to zero and any multiple of \[\pi \] gives sine equal to zero. The given function \[\cos \left( { - \dfrac{\pi }{2}} \right)\] is in the form of \[\cos \left( {\left( 0 \right)\pi - \dfrac{\pi }{2}} \right)\] , thus the cosine equals to zero.