Chapter 5.5, Problem 60E

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 $\tan u=\frac{-5}{12}$ and $\frac{3\pi }{2}<u<2\pi$ .

Answer to Problem 60E

The exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ are $\frac{\sqrt{26}}{26}$ , $\frac{5\sqrt{26}}{26}$ and $-\frac{1}{5}$ , respectively.

Explanation of Solution

Given information:

  $\tan u=\frac{-5}{12}$ and $\frac{3\pi }{2}<u<2\pi$

Calculation:

Let the given value is $\tan u=\frac{-5}{12}$ . The angle is $u$ and it ranges from $\frac{3\pi }{2}$ to $2\pi$ .

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

  $\begin{array}{c}\cos u=\left(\frac{12}{\sqrt{{12}^2+{\left(-5\right)}^2}}\right)\\ =\frac{12}{\sqrt{169}}\\ =\frac{12}{13}\end{array}$

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

  $\begin{array}{c}\sin u=-\left(\frac{5}{\sqrt{{12}^2+{\left(-5\right)}^2}}\right)\\ =\frac{-5}{\sqrt{169}}\\ =\frac{-5}{13}\end{array}$

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.

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

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.

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

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.

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

Therefore, the exact values of the $\sin \frac{u}{2}$ , $\cos \frac{u}{2}$ and $\tan \frac{u}{2}$ are $\frac{\sqrt{26}}{26}$ , $\frac{5\sqrt{26}}{26}$ and $-\frac{1}{5}$ , respectively.