Chapter 5, Problem 110RE

To determine

Tocalculate:The exact values of $\sin \left(\frac{u}{2}\right)$ , $\cos \left(\frac{u}{2}\right)$ and $\tan \left(\frac{u}{2}\right)$ .

Answer to Problem 110RE

The required values of $\sin \left(\frac{u}{2}\right)$ , $\cos \left(\frac{u}{2}\right)$ and $\tan \left(\frac{u}{2}\right)$ are 0.894, 0.447 and 2 respectively.

Explanation of Solution

Given information:

The identity: $\cos u=-\frac{3}{5}$ , $\pi <u<\frac{3\pi }{2}$

Formula used:

The half-angle formulas for trigonometric identities are shown below:

  $\begin{array}{l}\sin \left(\frac{u}{2}\right)=\pm \sqrt{\frac{1-\cos u}{2}}\\ \cos \left(\frac{u}{2}\right)=\pm \sqrt{\frac{1+\cos u}{2}}\\ \tan \left(\frac{u}{2}\right)=\frac{1-\cos u}{\sin u}\end{array}$

Pythagoras theorem:

For a right-angled triangle, the square of the hypotenuse and the sum of the squares of the base and the perpendicular are equal to each other that is,

  $\begin{array}{c}{\left(\text{hypotenuse}\right)}^2={\left(\text{perpendicular}\right)}^2+{\left(\text{base}\right)}^2\\ h^2=p^2+b^2\end{array}$

Calculation:

It is known that,

  $\sin \theta =\frac{\text{perpenidular}}{\text{hypotenuse}}=\frac{p}{h}$ and $\cos \theta =\frac{\text{base}}{\text{hypotenuse}}=\frac{b}{h}$

Here, $\cos u=-\frac{3}{5}=\frac{b}{h}$

This means, $b=-3$ and $h=5$

And, $\sin u=\frac{p}{h}$

To find the perpendicular that is, p; use the Pythagoras theorem $h^2=p^2+b^2$ .

Substitute $-3$ for $b$ and 5 for $h$ in the above formula and simplify for $p$ .

  $\begin{array}{c}{\left(5\right)}^2=p^2+{\left(-3\right)}^2\\ p^2=25-9\\ p^2=16\\ p=\pm 4\end{array}$

Since, sides cannot be negative; so, $p=4$

Substitute 4 for $p$ and 5 for $h$ in the formula $\sin u=\frac{p}{h}$ .

  $\sin u=\frac{4}{5}$

To find the exact values, use the formulas $\sin \left(\frac{u}{2}\right)=\pm \sqrt{\frac{1-\cos u}{2}}$ ,

  $\cos \left(\frac{u}{2}\right)=\pm \sqrt{\frac{1+\cos u}{2}}$ and $\tan \left(\frac{u}{2}\right)=\frac{1-\cos u}{\sin u}$ .

Substitute $-\frac{3}{5}$ for $\cos u$ in the formula $\sin \left(\frac{u}{2}\right)=\sqrt{\frac{1-\cos u}{2}}$ and simplify.

  $\begin{array}{c}\sin \left(\frac{u}{2}\right)=\sqrt{\frac{1+\frac{3}{5}}{2}}\\ =\sqrt{\frac{\frac{5+3}{5}}{2}}\\ =\sqrt{\frac{8}{10}}\\ \approx 0.894\end{array}$

Substitute $-\frac{3}{5}$ for $\cos u$ in the formula $\cos \left(\frac{u}{2}\right)=\sqrt{\frac{1+\cos u}{2}}$ and simplify.

  $\begin{array}{c}\cos \left(\frac{u}{2}\right)=\sqrt{\frac{1-\frac{3}{5}}{2}}\\ =\sqrt{\frac{\frac{5-3}{5}}{2}}\\ =\sqrt{\frac{2}{10}}\\ \approx 0.447\end{array}$

Substitute $-\frac{3}{5}$ for $\cos u$ and $\frac{4}{5}$ for $\sin u$ in the formula $\tan \left(\frac{u}{2}\right)=\frac{1-\cos u}{\sin u}$ and simplify.

  $\begin{array}{c}\tan \left(\frac{u}{2}\right)=\frac{1+\frac{3}{5}}{\frac{4}{5}}\\ =\frac{\frac{5+3}{\cancel{5}}}{\frac{4}{\cancel{5}}}\\ =\frac{8}{4}\\ =2\end{array}$

Hence, the required values of $\sin \left(\frac{u}{2}\right)$ , $\cos \left(\frac{u}{2}\right)$ and $\tan \left(\frac{u}{2}\right)$ are 0.894, 0.447 and 2 respectively.