Chapter 5.5, Problem 58E

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 $\sin u=\frac{5}{13}$ and $\frac{\pi }{2}<u<\pi$ .

Answer to Problem 58E

The exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ are $\frac{5\sqrt{26}}{26}$ , $\frac{\sqrt{26}}{26}$ and $5$ , respectively.

Explanation of Solution

Given information:

  $\sin u=\frac{5}{13}$ and $\frac{\pi }{2}<u<\pi$

Calculation:

Let the given value is $\sin u=\frac{5}{13}$ . The angle is $u$ and it ranges from $\frac{\pi }{2}$ to $\pi$ .

The value of $\cos u$ is calculated as,

  $\begin{array}{c}\cos u=-\left(\frac{\sqrt{{13}^2-5^2}}{13}\right)\\ =\frac{-12}{13}\end{array}$

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

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

Substitute $\frac{-12}{13}$ for $\cos u$ in the above equation.

  $\begin{array}{c}\sin \frac{u}{2}=\sqrt{\frac{1-\frac{-12}{13}}{2}}\\ =\sqrt{\frac{13+12}{26}}\\ =\sqrt{\frac{25}{26}}\\ =\frac{5\sqrt{26}}{26}\end{array}$

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

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

Substitute $\frac{-12}{13}$ for $\cos u$ in the above equation.

  $\begin{array}{c}\cos \frac{u}{2}=\sqrt{\frac{1+\frac{-12}{13}}{2}}\\ =\sqrt{\frac{13-12}{26}}\\ =\sqrt{\frac{1}{26}}\\ =\frac{\sqrt{26}}{26}\end{array}$

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

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

Substitute $\frac{5}{13}$ for $\sin \frac{u}{2}$ and $\frac{-12}{13}$ for $\cos \frac{u}{2}$ in the above equation.

  $\begin{array}{c}\tan \frac{u}{2}=\frac{1-\left(\frac{-12}{13}\right)}{\left(\frac{5}{13}\right)}\\ =5\end{array}$

Therefore, the exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ are $\frac{5\sqrt{26}}{26}$ , $\frac{\sqrt{26}}{26}$ and $5$ , respectively.