Question

The value of \[\sin \theta +\cos ({{90}^{\circ }}+\theta )+\sin ({{180}^{\circ }}-\theta )+\sin ({{180}^{\circ }}+\theta )\] is
A. -1
B. 0
C. \[\dfrac{1}{2}\]
D. 1

Answer 1

HINT:-Before solving this question, we must know about the relations that are there among the different values given by sin and cos functions on having angles that differ by multiples of \[{{90}^{\circ }}\] .

Complete step by step answer:
These relations are mentioned below
\[\begin{align}
  & \sin ({{90}^{\circ }}-\theta )=\cos \theta \\
 & \sin ({{90}^{\circ }}+\theta )=\cos \theta \\
 & \sin ({{180}^{\circ }}-\theta )=\sin \theta \\
 & \sin ({{180}^{\circ }}+\theta )=-\sin \theta \\
 & \cos ({{90}^{\circ }}-\theta )=\sin \theta \\
 & \cos ({{90}^{\circ }}+\theta )=-\sin \theta \\
 & \cos ({{180}^{\circ }}-\theta )=-\cos \theta \\
 & \cos ({{180}^{\circ }}+\theta )=-\cos \theta \\
\end{align}\]
Now, using these expressions and relations of sin and cos function, we will be able to solve the asked question.
As mentioned in the question, we have to evaluate the expression that is given in the question.
Now, on using the relations that are mentioned in the hint, we can write the expression as follows
\[\begin{align}
  & =\sin \theta +\cos ({{90}^{\circ }}+\theta )+\sin ({{180}^{\circ }}-\theta )+\sin ({{180}^{\circ }}+\theta ) \\
 & =\sin \theta +\left( -\sin \theta \right)+\sin \theta +\left( -\sin \theta \right)=0 \\
\end{align}\]
(Using the relations that are given in the hint)
Hence, we get the result of the expression to be 0.
NOTE:-
The students can make an error if they don’t know about the relations that are mentioned in the hint as they are very crucial to solve the problem and get to the solution.
Without knowing the relations one can never get to the correct answer.