Chapter 8.4, Problem 1QR

Solution

To find: The value of $\cos 2\theta$ .

The value of $\cos 2\theta$ is .

Given Information:

The given identity is $\cot 2\theta =\frac{5}{12}$ , $0<\theta <\frac{\pi }{2}$ .

Calculation:

Since, $\cot \theta =\frac{\text{Adjacent}}{\text{Opposite}}$ . Apply Pythagoras theorem to determine the value of the hypotenuse.

From the given identity, the value of adjacent is 5 and the value of opposite is 12.

  ${\text{Hypotenuse}}^2={\text{Adjacent}}^2+{\text{Opposite}}^2$

Substitute the values in the above equation.

  $\begin{array}{c}{\text{Hypotenuse}}^2={\text{5}}^2+{\text{12}}^2\\ =25+144\\ =169\end{array}$

Further, simplify.

  $\begin{array}{c}\text{Hypotenuse}=\sqrt{169}\\ =\pm 13\end{array}$

Since, $0<\theta <\frac{\pi }{2}$ . So, $0<2\theta <\pi$ and the value of $\cot$ is positive in the first quadrant. So, the value of $\cot 2\theta$ would be greater than 0. Consider the positive value of the hypotenuse.

Calculate the value of $\cos 2\theta$ .

  $\cos 2\theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}$

Substitute the values in the above equation.

  $\cos 2\theta =\frac{5}{13}$

Therefore, the value of $\cos 2\theta$ is $\frac{5}{13}$ .