Chapter 12.3, Problem 76WE

To determine

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

Answer to Problem 76WE

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

Explanation of Solution

Given information :

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

Here we shall use a trigonometric identity,

  $\tan \left(a-b\right)=\frac{\tan a-\tan b}{1+\tan a\tan b}$

Now we shall solve,

  $\tan \left({180}^{\circ }-\theta \right)=\frac{\tan {180}^{\circ }-\tan \theta }{1+\tan {180}^{\circ }\tan \theta }$

Value of $\tan {180}^{\circ }=0$

  $\frac{\tan {180}^{\circ }-\tan \theta }{1+\tan {180}^{\circ }\tan \theta }=\frac{0-\tan \theta }{1+0}=-\tan \theta$

Now we know that $\tan \theta =\frac{\sin \theta }{\cos \theta }$

Hence we get,

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