Chapter 6, Problem 22CLT

To determine

  $\sin \left(\frac{u}{2}\right)$ angle ulies in Quadrant I.

Answer to Problem 22CLT

The value of $\sin \left(\frac{u}{2}\right)=\frac{1}{\sqrt{5}}$

Explanation of Solution

Given info:

  $\begin{array}{l}\tan u=\frac{4}{3}\\ uliesinQuadrantI\end{array}$

Formula used:

  $\sin \left(\frac{u}{2}\right)=\sqrt{\frac{1-\cos u}{2}}$

Calculation:

Given that

  $\begin{array}{l}\tan u=\frac{4}{3}\\ uliesinQuadrantI\end{array}$

We have

  $\begin{array}{l}\sin \left(\frac{u}{2}\right)=\sqrt{\frac{1-\cos u}{2}}\\ \tan u=\frac{4}{3}\\ y=4,x=3\\ r=\sqrt{3^2+4^2}=5\\ \cos u=\frac{x}{r}=\frac{3}{5}\\ \sin \left(\frac{u}{2}\right)=\sqrt{\frac{1-\frac{3}{5}}{2}}=\sqrt{\frac{\frac{5-3}{5}}{2}}\\ \sin \left(\frac{u}{2}\right)=\frac{1}{\sqrt{5}}\end{array}$

The value of $\sin \left(\frac{u}{2}\right)=\frac{1}{\sqrt{5}}$

Conclusion:

Thus,the value of $\sin \left(\frac{u}{2}\right)=\frac{1}{\sqrt{5}}$