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Hello! Would like help on parts b and c. Thank you!

Hello! Would like help on parts b and c. Thank you!-example-1

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The function given describes the difference in lenght after theta seconds:


f(\theta)=2sin\theta+\sqrt[\placeholder{⬚}]{2}

Part A:

When the pogo stick's spring is equal to its no compressed lenght is when the difference is equal to 0.

therefore, We have to do f(theta)=0.


\begin{gathered} f(\theta)=0 \\ 2sin\theta+\sqrt[\placeholder{⬚}]{2}=0 \\ \end{gathered}

Solving for theta:


\begin{gathered} 2sin\theta=-\sqrt[\placeholder{⬚}]{2} \\ sin(\theta)=\frac{-\sqrt[\placeholder{⬚}]{2}}{2} \\ \theta=sin^(-1)(\frac{-\sqrt[\placeholder{⬚}]{2}}{2})=-(1)/(4)\pi \end{gathered}

Part B:

If theta is equal to 2*theta in the interval [0,2pi)


f(2\theta)=2sin2\theta+\sqrt[\placeholder{⬚}]{2}

Evaluating in 0 and 2pi:


\begin{gathered} f(0)=2s\imaginaryI n2*0+√(2)=2*0+\sqrt[\placeholder{⬚}]{2}=\sqrt[\placeholder{⬚}]{2} \\ f(2(2\pi))=2s\imaginaryI n2*2\pi+√(2)=2*sin4\pi+\sqrt[\placeholder{⬚}]{2}=2*0+\sqrt[\placeholder{⬚}]{2} \\ f(2*(\pi)/(2))=2s\imaginaryI n2*(\pi)/(2)+√(2)=2*sin\pi+\sqrt[\placeholder{⬚}]{2}=2*0+\sqrt[\placeholder{⬚}]{2} \end{gathered}

Is the same than the original function because the sin of (2n*pi) is going to be always 0.

Part C:

Given the function for the another pogo:


g(\theta)=1-cos^2\theta+\sqrt[\placeholder{⬚}]{2}

As you can see, by trigonometric identity, we can rewrite the function g of the following way:


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

Substituing:


g(x)=sin^2+\sqrt[\placeholder{⬚}]{2}

Matching g and f:


\begin{gathered} sin^2\theta+\sqrt[\placeholder{⬚}]{2}=2sin\theta+\sqrt[\placeholder{⬚}]{2} \\ sin^2\theta=2sin\theta \\ sin\theta=2 \end{gathered}

Both length will be equal when the sin of theta is equal to 2.

User Nilsocket
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