Solve each equation on the interval [0, 2𝝅). Use exact values...

Question

Solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + cos x = 0

Accepted Answer

Hello, everyone. We are asked to apply the most suitable technique to determine the solution of the following equation. Within the interval. 0 to 2 pi including zero. We are going to employ exact values whenever feasible or approximations accurate to four significant figures. The equation we are given is 10, multiplied by the cosine of X plus five radical two multiplied by the sine of two X equals zero. Our answer choices are a no solution. B X equals the set pi divided by 27 pi divided by 63 pi divided by 2, 11 pi divided by six C X equals the solution. And set pi divided by 24 pi divided by three, three pi divided by 25 pi divided by three D X equals the set pi divided by 25 pi divided by 43 pi divided by 27 pi divided by four. Looking at the equation we are given, we have 10 multiplied by the cosine of X plus five radical two multiplied by the sign of two X equal to zero. Thinking back to my identities. I recall that the sign of two acts is equivalent to two sine X cosine X. And I think that might help us out if we substitute that in for the sign of two X. So if we do so we have 10 multiplied by the cosine of X plus five radical two multiplied by two sin of X cosine of X equal to zero. I'm gonna multiply together five radical two and two sine X cosine X. So we have 10 cosine of X plus radical two multiplied by the sign of X cosine of X equal to zero. I put 10 radical two in parentheses. Just to show that the radical is not over the sine X cosine X. Next, I notice both terms have a 10 and a cosine X. So I'm gonna factor that out as the greatest common factor. So I have 10 cosine X multiplied by the parentheses. One plus radical two sine X closed parentheses equals zero. Thinking back to algebra, I know that when I have factors equal to zero, I can set each factor equal to zero and solve. So I'm gonna start with 10 cosine of X and I'm gonna make that equal to zero. I divide both sides by 10. We get the cosine of X equals zero. Looking at a unit circle, I can find that X has to be the values pi divided by two and three pi divided by two for cosine to equal zero. So cosine of X is zero at pi divided by two and three pi divided by two. So this is going to be part of our answer, part of our X solution set. We don't have all of it done just yet. Next, I'm going to set the other factor one plus radical two multiplied by the sin of X equal to zero. Solving this for X I subtract one from both sides. I have radical two multiplied by the sin of X equals negative one, dividing both sides by radical two, I get the sign of X equals negative one divided by radical two. And when I rationalize that, that's gonna be negative radical two divided by two. So again, referring back to my unit circle to see when sine equals negative radical two divided by two. And we find that that occurs at five pi divided by four and seven pi divided by four. So those are our two other angles that could be part of this solution. So now checking my answer choices, I'm looking for pi divided by two. So that's all BC and D um five pi divided by four, that one only occurs in D. So let's double check the rest three pi divided by two and seven pi divided by four. Both are there. So answer choice D is our answer. Have a nice day.

Solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + cos x = 0

Key Concepts

Double-Angle Identity for Sine

Solving Trigonometric Equations

Interval Restriction and Exact Values

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