Chapter 5, Problem 14PS

(a)

To determine

To write: a sum formula for $\sin \left(u+v+w\right)$ .

Answer to Problem 14PS

  $\sin \left(u+v+w\right)=\sin u\cos v\cos w+\cos u\sin v\cos w+\cos u\cos v\sin w-\sin u\sin v\sin w$

Explanation of Solution

Given information:

Given equation

  $\sin \left(u+v+w\right)$ .

Calculation:

The sum formulas for sine and cosine function are given as follows:

  $\begin{array}{l}\sin \left(u+v\right)=\sin u\cos v+\cos u\sin v\\ \cos \left(u+v\right)=\cos u\cos v-\sin u\sin v\end{array}$

The given expression sum formula for $u,v$ and $w$ can be simplified using the sum formulas given above.

  $\begin{array}{c}\sin \left(u+v+w\right)=\sin u\cos \left(v+w\right)+\cos u\sin \left(v+w\right)\\ =\sin u\left(\cos v\cos w-\sin v\sin w\right)+\cos u\left(\sin v\cos w+coev\sin w\right)\\ =\sin u\cos v\cos w-\sin u\sin v\sin w+\cos u\sin v\cos w+\cos w\cos v\sin w\\ =\sin u\cos v\cos w+\cos u\sin v\cos w+\cos u\cos v\sin w-\sin u\sin v\sin w\end{array}$

(b)

To determine

To write: a sum formula for $\tan \left(u+v+w\right)$ .

Answer to Problem 14PS

  $\tan \left(u+v+w\right)=\frac{\tan u+\tan v+\tan w-\tan u\tan v\tan w}{1-\tan u\tan v-\tan u\tan w-\tan v\tan w}$

Explanation of Solution

Given information:

Given function.

  $\tan \left(u+v+w\right)$

Calculation:

The sum formulas for sine and cosine function are given as follows:

  $\tan \left(u+v\right)=\frac{\tan u+\tan v}{1-\tan u\tan v}$

The given expression sum formula for $u,v$ and $w$ can be simplified using the sum formulas given above.

  $\begin{array}{c}\tan \left(u+v+w\right)=\frac{\tan \left(u+v\right)+\tan w}{1-\tan \left(u+v\right)\tan w}\\ =\frac{\frac{\tan u+\tan v}{1-\tan u\tan v}+\tan w}{1-\frac{\tan u+\tan v}{1-\tan u\tan v}\tan w}\\ =\frac{\frac{\tan u+\tan v+\tan w-\tan u\tan v\tan w}{1-\tan u\tan v}}{\frac{1-\tan u\tan v-\tan u\tan v-\tan v\tan w}{1-\tan u\tan v}}\\ =\frac{\tan u+\tan v+\tan w-\tan u\tan v\tan w}{1-\tan u\tan v-\tan u\tan w-\tan v\tan w}\end{array}$