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10 votes
10 votes
Find all exact solutions
2cos^2x=sinx+1
Please show work thank you!

User Teubanks
by
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1 Answer

10 votes
10 votes

Answer:


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Explanation:

We are given the equation:


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And we want to find all solutions for x.

First, we should put the equation into terms of only one trigonometric ratio.

Since we are given cos²(x), we can turn this into sine. Recall the Pythagorean Identity which states:


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Therefore:


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By substitution:


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Distribute:


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Isolate the equation:


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We can factor:


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Zero Product Property:


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Solve for each case:


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We can use the unit circle.

sin(x) = 1/2 for every π/6 and 5π/6. So, it will continue every 2π.

sin(x) = -1 every 3π/2. And this will also continue every 2π.

Hence, our solutions are:


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Note:

If you only need the solutions within the interval [0, 2π), then it is:


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User Kazenorin
by
2.6k points