Chapter 5.5, Problem 76E

(a)

To determine

Verify the double angle formula using given figure.

Answer to Problem 76E

It is verified.

Explanation of Solution

Given:

Consider the below figure

  

  $sin2u=2sinucosu$

From the above figure $\sin 2u=1,\cos 2u=0,\sin u=\cos u=\frac{1}{\sqrt{2}}$

Now, consider $2\sin u\cos u$

  $\begin{array}{cc}2\sin u\cos u& =2\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right)\\ & =\frac{2}{2}\\ & =1\\ & =\sin 2u\\ & \end{array}$

Hence $sin2u=2sinucosu$

(b)

To determine

Verify the double angle formula using given figure.

Answer to Problem 76E

It is verified.

Explanation of Solution

Given:

Consider the below figure

  

From the above figure $\sin 2u=1,\cos 2u=0,\sin u=\cos u=\frac{1}{\sqrt{2}}$

Now consider ${\cos }^{2}u-{\sin }^{2}u$

  $\begin{array}{cc}{\cos }^{2}u-{\sin }^{2}u& =0(since\sin u=\cos u)\\ & =\cos 2u\\ \text{Hence}& \cos 2u={\cos }^{2}u-{\sin }^{2}u\end{array}$