Chapter 5.5, Problem 64E

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 $\sec u=\frac{7}{2}$ and $0<u<\frac{\pi }{2}$ .

Answer to Problem 64E

The exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ are $\sqrt{\frac{5}{14}}$ , $\frac{3\sqrt{14}}{14}$ and $\frac{\sqrt{5}}{3}$ , respectively.

Explanation of Solution

Given information:

  $\sec u=\frac{7}{2}$ and $0<u<\frac{\pi }{2}$ .

Calculation:

Let the given value is $\sec u=\frac{7}{2}$ . The angle is $u$ and it ranges from $0$ to $\frac{\pi }{2}$ .

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

  $\cos u=\frac{2}{7}$

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

  $\begin{array}{c}\sin u=\frac{\sqrt{7^2-2^2}}{7}\\ =\frac{\sqrt{45}}{7}\\ =\frac{3\sqrt{5}}{7}\end{array}$

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

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

Substitute $\frac{2}{7}$ for $\cos u$ in the above equation.

  $\begin{array}{c}\sin \frac{u}{2}=\sqrt{\frac{1-\frac{2}{7}}{2}}\\ =\sqrt{\frac{7-2}{14}}\\ =\sqrt{\frac{5}{14}}\end{array}$

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

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

Substitute $\frac{2}{7}$ for $\cos u$ in the above equation.

  $\begin{array}{c}\cos \frac{u}{2}=\sqrt{\frac{1+\frac{2}{7}}{2}}\\ =\sqrt{\frac{9}{14}}\\ =\frac{3\sqrt{14}}{14}\end{array}$

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 $\frac{3\sqrt{5}}{7}$ for $\sin \frac{u}{2}$ and $\frac{2}{7}$ for $\cos \frac{u}{2}$ in the above equation.

  $\begin{array}{c}\tan \frac{u}{2}=\frac{1-\left(\frac{2}{7}\right)}{\left(\frac{3\sqrt{5}}{7}\right)}\\ =\frac{5}{3\sqrt{5}}\\ =\frac{\sqrt{5}}{3}\end{array}$

Therefore, the exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ are $\sqrt{\frac{5}{14}}$ , $\frac{3\sqrt{14}}{14}$ and $\frac{\sqrt{5}}{3}$ , respectively.