Chapter 12.3, Problem 72WE

To determine

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

Answer to Problem 72WE

  $\sin \left({180}^{\circ }-\theta \right)=\sin \theta$

Explanation of Solution

Given information :

  $\sin \left({180}^{\circ }-\theta \right)$

Here we shall use a trigonometric identity,

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

Now we shall solve,

  $\sin \left({180}^{\circ }-\theta \right)=\sin {180}^{\circ }\cos \theta -\cos {180}^{\circ }\sin \theta$

  $\left(\sin {180}^{\circ }=0,\cos {180}^{\circ }=-1\right)$

  $\sin {180}^{\circ }\cos \theta +\cos {180}^{\circ }\sin \theta =0-\left(-1\right)\sin \theta$

Hence we can say that $\sin \left({180}^{\circ }-\theta \right)=\sin \theta$

Also the given angle is in II quadrant hence it will be positive as sine is positive in II quadrant.