Solving Trigonometric Equations

Question

For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.

sin² θ = cos² θ

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Solve for theta in the interval, square brackets, 0.2 pi if 3 multiplied by the cosine squared of theta equals 3 divided by 2. Or 3 halves. Awesome. So it appears for this particular problem we're asked to solve for theta within the specific interval if 3 multiplied by cosine square to theta equals 3 halves. Awesome. And it's important to note that our interval, which is square brackets 0.2, is within square brackets, which means it is a closed interval. That's very important to take note of. So, with that said, Let's read off our multiple choice answers now that we know that we're ultimately trying to solve for theta and see which one must be our final answer. And once again, let us know that they all state that theta is equal to some values. So, A is pi divided by 33 pi divided by 4. 5 pi divided by 3, and 7 pi divided by 4. B is pi divided by 43 pi divided by 45 pi divided by 4, and 7 pi divided by 4. C is pi divided by 23 pi divided by 45 pi divided by 47 pi divided by 4. And finally, D is pi divided by 43 pi divided by 25 pi divided by 4, and 7 pi divided by 4. OK. So first off, we are trying to solve for theta within the interval square brackets 0.2 pi, and we're given the equation 3 multiplied by cosine square to theta equals 3 divided by 2 or 3 halves. So, in order to do that, we must first simplify the equation. So let us divide both sides by 3 to isolate cosine square to theta. So when we do that, we will find that cosine squared. Of theta is going to be equal to 3 halves divided by 3, which is going to be equal to 1/2. Our second step, so that was our first step. Our second step is we need to solve for cosine theta. So in order to solve for cosine theta, we need to take the square root of both sides, so we need to take cosine of theta is going to be equal to plus or minus the square root of 1/2, which is going to be equal to + or minus the square root of 2 divided by 2. Fantastic. So now our third step is we need to find theta within our intervals. We need to find theta in our closed interval of 0.2 pi. Fantastic. So with that said, Our next course of action, now that we're dealing with a closed interval, it's very important that this is that we know it's a closed interval. We need to recall and note that the cosine of an angle is square rooted 2 divided by 2, or negative square rooted 2 divided by 2 at specific standard angles within our closed interval, and these angles will be Note that cosine of theta is equal to 22 divided by 2. Let us note that theta equals pi divided by 4, and this is in the first quadrant. Let us also note that theta equals 7 pi divided by 4, and this will be in the 4th quadrant. And focusing our attention now on cosine of theta equals negative 22 divided by 2, that will mean that theta is going to be equal to 3 pi divided by 4, and this will be in the 2nd quadrant. And let us also note that theta equals 5 pi divided by 4, and once again this will be in the 3rd quadrant. Fantastic. So now we need to list all of our solutions for theta within our closed interval. And when we do that, since we've already determined all of our final answers, so let's quickly box them so we know that. Theta is equal to pi divided by 4. We know that theta is equal to 7 pi divided by 4. We know that theta is going to be equal to 3 pi divided by 4, and theta is going to be equal to 5 pi divided by 4. So that means, without a doubt, our multiple choice answer that has to be correct, has to be multiple choice answer letter B, which once again is theta equals pi divided by 4. 3 divided by 45 divided by 4, and 7 pi divided by 4. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Solving Trigonometric Equations

For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.

sin² θ = cos² θ

Key Concepts

Trigonometric Identities