Chapter 7, Problem 66RE

To determine

To find: The exact value of the expression $\tan \frac{y}{2}$ if $\sec x=\frac{3}{2}$ and $\csc y=3$ where $x$ and $y$ are in Quadrant I.

Answer to Problem 66RE

The value of the expression $\tan \frac{y}{2}$ is $3-2\sqrt{2}$ .

Explanation of Solution

Given information:

The given expression is $\tan \frac{y}{2}$ . The values are $\sec x=\frac{3}{2}$ and $\csc y=3$ .

Calculation:

The values are $\sec x=\frac{3}{2}$ and $\csc y=3$ . Since $x$ and $y$ are in Quadrant I. All values will be positive.

  $\begin{array}{l}\sec x=\frac{3}{2}\\ \cos x=\frac{2}{3}\end{array}$

Apply Pythagoras theorem to calculate the value of perpendicular as $\sqrt{5}$ . So, the value of $\sin x$ will be $\frac{\sqrt{5}}{3}$ .

  $\begin{array}{c}\csc y=3\\ \sin y=\frac{1}{3}\end{array}$

Apply Pythagoras theorem to calculate the value of base as $\sqrt{8}$ . So, the value of $\cos y$ will be $\frac{\sqrt{8}}{3}$ .

Solve the expression using sum formula.

  $\begin{array}{c}\tan \frac{y}{2}=\frac{1-\cos y}{\sin y}\\ =\frac{1-\frac{\sqrt{8}}{3}}{\frac{1}{3}}\\ =3-\sqrt{8}\\ =3-2\sqrt{2}\end{array}$

Therefore, the value of the expression $\tan \frac{y}{2}$ is $3-2\sqrt{2}$ .