Solve each equation in x over the interval [0, 2π) and each eq...

Question

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

√2 cos 2θ = -1

Accepted Answer

Hello, everyone. We are asked to find the exact solutions of the equation considering the interval from 0 to 2 pi including zero. We are given the equation two cosine four X equals one. We have four answer choices which are different sets of values and have different amounts of values in each set to begin with. I'm going to rewrite my equation. So I have two cosign four X equals one. I'm going to divide both sides by two to try to isolate the cosine. And I get the cosine of four X equals one half from here. Since I'm trying to find the value for X, I'm going to use the inverse cosine of one half and say that it equals four X. So I want to find using my unit circle. What is the inverse cosine of one half? Well, it has two values. It could be pi divided by three or five pi divided by three. So this means we will have two different equations to set up to find our solutions. First equation will be that our four X will equal pi divided by three. But we're also gonna add plus two pi N because N is the number of rotations that we may have made around this circle. We don't really know. So we have to put it in there. So I'm divide everything by four to get X by itself, I will get X equals pi divided by 12 plus pi N divided by two. So now let's plug some values in for N thinking about pi divided by two rotations, I could make three rotations in the interval we're given. So we want to check when N equals 01, two or three. So starting with, when N equals zero, plugging that into this, we have X equals pi divided by 12 plus pi multiplied by zero, divided by two. Anything multiplied by zero is zero, divided by two is still zero. This means one of our solutions is pi divided by 12. Next, when N equals one, we'll have X equals pi divided by 12 plus pi divided by two. Because when we multiply it by one, it stays the same, making a common denominator. I'll have pi divided by 12 plus six, pi divided by 12. There'll be seven pi divided by 12. There's our next solution. Next. When N equals two, we'll have X equals pi divided by 12 plus. We said it was two. So it'll be two pi divided by two which does reduce just to pie. But since I need a common denominator, I'm gonna make it have a common denominator of 12. So we'll have 12 pie plus 12 pi divided by 12. So this will be 13 pi divided by 12. And now when N equals three, X will equal pi divided by 12 plus three pi divided by two. Again, I need a common denominator. So I'll have pi divided by 12 plus. In this case, it'll be 18 pi divided by 12. She'll give us a total of 19 pi divided by 12. All right, we have some of our answers done. We do still have another angle to take into consideration. So the next equation we set up will have four, X equals five pi divided by three plus two pie, N because again, it could land at a different terminating side of five pi divided by three. So to get the X value, I divide everything by four. So I have the equation X equals it will be five pi divided by 12 plus pi N divided by two. So again, I could still make the same number of rotations. So I need to check when N is 012 and three. So beginning with, when N is zero, love X equals five pi divided by 12 plus pi multiplied by zero, divided by two. Anything multiplied by zero is zero. So this solution is five pi divided by 12. Next, when N equals one, we'll have X equals five pi divided by 12 plus pi divided by two. And again, I need a common denominator. So that will be 12. So I have five pi divided by 12 plus six pi divided by 12. And that'll give us a solution of 11 pi divided by 12. Next, when N equals two X will equal five pi divided by 12 plus two pi divided by two. Again, we could simplify that, but we do need a common denominator. So we might as well just make our common denominator. So we'll have five pi divided by 12 plus 12 pi divided by 12. That would be 17 pi divided by 12. And the last one when N equals three X equals five pi divided by 12 plus three pi divided by two, making common denominator of 12. I'll have five pi divided by 12 plus 18 pi divided by 12 and this will give me 23 P divided by 12. So we have eight solutions. Our solutions are pi divided by 12, 5 pi divided by 12, 7 pi divided by 1211 pi divided by 1213 pi divided by 1217 pi divided by 1219 pi divided by 12 and 23 pi divided by 12. And that is answer choice. A have a nice day.

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

√2 cos 2θ = -1

Key Concepts

Double-Angle Trigonometric Functions

Solving Trigonometric Equations

Interval and Exact Solutions in Radians and Degrees

Watch next