Question

Evaluate the given trigonometric function $\cos {90^\circ }\sin {\theta ^\circ } - \sin {\theta ^\circ }\cos {90^\circ }$

Answer 1

Hint: In this particular question use the concept that the value of the $\cos {90^\circ }$ is always zero as shown in the below graph, so use this property in the given trigonometric equation to reach the solution of the question.

Complete step-by-step answer:
Given trigonometric equation:
$\cos {90^\circ }\sin {\theta ^\circ } - \sin {\theta ^\circ }\cos {90^\circ }$
As we all know that at 90 degrees (i.e. at $\dfrac{\pi }{2}$ radian) the graph of the cosine cut the x-axis as shown in the above diagram so the value of the $\cos {90^\circ } = 0$, so use this property in the given trigonometric equation we have,
$ \Rightarrow \left( 0 \right)\sin {\theta ^\circ } - \sin {\theta ^\circ }\left( 0 \right)$
Now as we know that anything multiplied by zero is zero so use this property in the above equation we have,
$ \Rightarrow 0 - 0 = 0$
So the value of the given trigonometric equation is 0.
Hence, $\cos {90^\circ }\sin {\theta ^\circ } - \sin {\theta ^\circ }\cos {90^\circ }$ = 0
So this is the required answer.

Note: Whenever we face such types of questions the key concept is the graph of the cosine which is shown above in the diagram, so from this graph we clearly say that the graph of the cosine cut the x-axis at $90^\circ$, $270^\circ$ and at this point the value of the cosine is zero, so this is the basis of the solution, so substitute zero (0) in place of $\cos {90^\circ}$ in the given trigonometric equation as above and simplify we will get the required answer.