Chapter 5.4, Problem 1E

To determine

To fill:To fill in the blanks of the given statements.

Answer to Problem 1E

The final answer is $\sin \left(u-v\right)=\sin u.cosv-\cos u.\sin v$

Explanation of Solution

Given information:we have given that $\sin \left(u-v\right)$

From sum and difference formula we know that

  $\sin \left(u+v\right)=\sin u.cosv+\cos u.\sin v$

Now, we write

  $\sin \left(u+\left(-v\right)\right)=\sin u.cos\left(-v\right)+\cos u.\sin \left(-v\right)$ …..(1)

As cosine function is an even function and sine function is an odd function.

So must have

  $\cos \left(-\theta \right)=\cos \theta and\sin \left(-\theta \right)=-\sin \theta$ …..(2)

Using (2) in (1) we get

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

Finally we have

  $\sin \left(u-v\right)=\sin u.cosv-\cos u.\sin v$