Chapter 5.5, Problem 59E

To determine

To find:The exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ using half-angle formula when $\cos u=\frac{7}{25}$ and $0<u<\frac{\pi }{2}$ .

Answer to Problem 59E

The exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ are $\frac{3}{5}$ , $\frac{4}{5}$ and $\frac{3}{4}$ , respectively.

Explanation of Solution

Given information:

  $\cos u=\frac{7}{25}$ and $0<u<\frac{\pi }{2}$

Calculation:

Let the given value is $\cos u=\frac{7}{25}$ . The angle is $u$ and it ranges from 0 to $\frac{\pi }{2}$ .So, the value of $u$ and $\frac{u}{2}$ will lie in the first quadrant and all the values will be positive.

Write the expression to calculate the value of $\sin \frac{u}{2}$ .

  $\sin \frac{u}{2}=\sqrt{\frac{1-\cos u}{2}}$

Substitute $\frac{7}{25}$ for $\cos u$ in the above equation.

  $\begin{array}{c}\sin \frac{u}{2}=\sqrt{\frac{1-\frac{7}{25}}{2}}\\ =\sqrt{\frac{25-7}{25\times 2}}\\ =\sqrt{\frac{18}{50}}\\ =\frac{3}{5}\end{array}$

Write the expression to calculate the value of $\cos \frac{u}{2}$ .

  $\cos \frac{u}{2}=\sqrt{\frac{1+\cos u}{2}}$

Substitute $\frac{7}{25}$ for $\cos u$ in the above equation.

  $\begin{array}{c}\cos \frac{u}{2}=\sqrt{\frac{1+\frac{7}{25}}{2}}\\ =\sqrt{\frac{25+7}{25\times 2}}\\ =\sqrt{\frac{32}{50}}\\ =\frac{4}{5}\end{array}$

Similarly, write the expression to calculate the value of $\tan \frac{u}{2}$ .

  $\tan \frac{u}{2}=\frac{\sin \frac{u}{2}}{\cos \frac{u}{2}}$

Substitute $\frac{3}{5}$ for $\sin \frac{u}{2}$ and $\frac{4}{5}$ for $\cos \frac{u}{2}$ in the above equation.

  $\begin{array}{c}\tan \frac{u}{2}=\frac{\left(\frac{3}{5}\right)}{\left(\frac{4}{5}\right)}\\ =\frac{3}{4}\end{array}$

Therefore, the exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ are $\frac{3}{5}$ , $\frac{4}{5}$ and $\frac{3}{4}$ , respectively.