Question

The value of \[\sin \left( {{360}^{\circ }}-\theta \right)\] is
a. \[-\sin \theta \]
b. \[-\cos \theta \]
c. \[\cos \theta \]
d. \[\sin \theta \]

Answer 1

Hint: Find the trigonometric ratios of function \[\theta \] and compare them to \[\left( 360-\theta \right)\]. \[\theta \] and \[\left( 360-\theta \right)\] are co – terminate angles i.e. they have a common terminal side. Thus get the value \[\sin \left( 360-\theta \right)\] from \[\sin \left( -\theta \right)\].

Complete step-by-step answer:

The trigonometric ratios of \[\left( 360-\theta \right)\], the terminal sides of co – terminal angles coincide, hence their trigonometric ratios are the same.
We know that,
\[\sin \left( -\theta \right)=-\sin \theta \]

Similarly, \[\sin \left( 360-\theta \right)=-\sin \theta \].
Thus we can clearly state that \[\left( 360-\theta \right)\] and \[\left( -\theta \right)\] are co – terminal sides. For example angles \[{{30}^{\circ }},-{{300}^{\circ }}\] and \[{{390}^{\circ }}\] are all terminal.
We can also state other examples like \[{{45}^{\circ }},{{405}^{\circ }}\] and \[-{{315}^{\circ }}\].

We can find the co – terminal angles of other functions.
\[\begin{align}
  & \sin \left( -\theta \right)=-\sin \theta \\
 & \cos \left( -\theta \right)=\cos \theta \\
 & \tan \left( -\theta \right)=-\tan \theta \\
 & \therefore \sin \left( 360-\theta \right)=-\sin \theta \\
 & \cos \left( 360-\theta \right)=\cos \theta \\
 & \tan \left( 360-\theta \right)=-\tan \theta \\
\end{align}\]

Thus we got the value of \[\sin \left( 360-\theta \right)=-\sin \theta \].
\[\therefore \] Option (a) is correct.

Note: If you know the trigonometric ratios of \[\theta \], then you can easily find \[\left( 360-\theta \right)\] of any function. Thus remember the functions so you can easily find the other functions values.