Chapter 11.1, Problem 7QR

To determine

To find: The equation relating $x$ and $y$ , when $x=\cos \theta$ and $y=\cos \left(2\theta \right)$ .

Answer to Problem 7QR

The relation between $x$ and $y$ is $2x^2-y^2=1$ .

Explanation of Solution

Given information:

  $\begin{array}{l}x=\cos \theta \\ y=\cos 2\theta \end{array}$

Formula used:

Trigonometric formula for angle $\theta$ .

  $\cos \left(2\theta \right)=2{\cos }^2\theta -1$

Calculation:

Substituting the value $x$ for $\cos \theta$ and $y$ for $\cos \left(2\theta \right)$ in the formula $\cos \left(2\theta \right)={\cos }^2\theta -1$ , we get

  $\begin{array}{c}\cos \left(2\theta \right)=2{\cos }^2\theta -1\\ y^2=2x^2-1\\ 2x^2-y^2=1\end{array}$

Conclusion:

Hence, the required relation between $x$ and $y$ is $2x^2-y^2=1$ .