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Solve the equation on the interval [0, 2\small \pi). Show all work. Do not use a calculator - use your unit circle!

Solve the equation on the interval [0, 2\small \pi). Show all work. Do not use a calculator-example-1
User Anish
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1 Answer

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SOLUTION

Write out the equation given


\cos ^2x+2\cos x-3=0

Let


\text{Cosx}=P

Then by substitution, we obtain the equation


p^2+2p-3=0

Solve the quadractic equation using factor method


\begin{gathered} p^2+3p-p-3=0 \\ p(p+3)-1(p+3)=0 \\ (p-1)(p+3)=0 \end{gathered}

Then we have


\begin{gathered} p-1=0,p+3=0 \\ \text{Then} \\ p=1,p=-3 \end{gathered}

Recall that


\cos x=p

Hence


\begin{gathered} \text{when p=1} \\ \cos x=1 \\ \text{Then } \\ x=\cos ^(-1)(1)=0 \\ \text{hence } \\ x=0 \end{gathered}

Similarly,


\begin{gathered} \text{When p=-3} \\ \cos x=-3 \\ x=\cos ^(-1)(-3) \\ x=no\text{ solution} \end{gathered}

Therefore x=0 is the only valid solution on the given interval [0,2π).

Answer; x=0

User Boonyongyang
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