Chapter 12.3, Problem 68WE

To determine

Write given in terms of $\sin \theta$ and $\cos \theta$ .

Answer to Problem 68WE

  $\sin \left(-\theta \right)=-\sin \theta$

Explanation of Solution

Given information :

  $\sin \left(-\theta \right)$

Here we shall use a trigonometric identity,

  $\sin \left(a-b\right)=\sin a\cos b-\cos a\sin b$

Now we shall put $\sin \left(-\theta \right)=\sin \left({360}^{\circ }-\theta \right)$

Now we know,

  $\sin \left({360}^{\circ }-\theta \right)=\sin {360}^{\circ }\cos \theta -\cos {360}^{\circ }\sin \theta$

  $\left(\sin {360}^{\circ }=0,\cos {360}^{\circ }=1\right)$

  $\sin {360}^{\circ }\cos \theta -\cos {360}^{\circ }\sin \theta =0-\sin \theta$

Hence we can say that $\sin \left(-\theta \right)=-\sin \theta$