Question

A toddler is jumping on another pogo stick whose length of their spring can be represented by the function g of theta equals 1 minus sine squared theta plus radical 3 period At what times are the springs from the original pogo stick and the toddler's pogo stick lengths equal?

Answer 1

Answer:

The springs from the original pogo stick and the toddler's pogo stick length are equal after 1 second and 0.9994 second.

Explanation:

The given functions are:

$\begin{gathered} f(\theta)=2\cos \theta+\sqrt[]{3} \\ g(\theta)=1-\sin ^2\theta+\sqrt[]{3} \end{gathered}$

The springs from the original pogo stick and the toddler's pogo stick length are equal when both functions coincide

That is;

$\begin{gathered} f(\theta)=g(\theta) \\ \Rightarrow2\cos \theta+\sqrt[]{3}=1-\sin ^2\theta+\sqrt[]{3} \end{gathered}$

Solving the equation, we have:

$\begin{gathered} 2\cos \theta+\sqrt[]{3}=1-\sin ^2\theta+\sqrt[]{3} \\ Subtract\sqrt[]{3}\text{ from both sides} \\ 2\cos \theta=1-\sin ^2\theta \end{gathered}$

Note the identity below:

$\begin{gathered} \cos ^2\theta+\sin ^2\theta=1 \\ \cos ^2\theta=1-\sin ^2\theta \end{gathered}$

This means

$\begin{gathered} 2\cos \theta=\cos ^2\theta \\ \cos ^2\theta-2\cos \theta=0 \\ \cos \theta(\cos \theta-2)=0 \\ \cos \theta=0 \\ \Rightarrow\theta=\cos ^{-1}(0)=1 \\ \\ OR \\ \cos \theta-2=0 \\ \cos \theta=2 \\ \theta=\cos ^{-1}(2)=0.9994 \end{gathered}$

The springs from the original pogo stick and the toddler's pogo stick length are equal after 1 second and 0.9994 second.