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

Question

cos 2θ + cos θ = 0

Accepted Answer

Hello there. Today we're gonna 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 the following trigontric equation for theta, where 0 is less than or equal to theta and theta is less than or equal to pi. Cosine of 2 theta minus cosine of theta is equal to 0. Fantastic. So it appears for this particular problem, we're asked to solve this trigontric equation for theta, given this specific range or interval. And we're told that the specific range is 0 is less than or equal to theta and theta is less than or equal to pi. So now that we know what we're ultimately trying to solve or we're trying to solve this trigonometric equation, let's read off our multiple choice sensors to see what our final answer might be, noting that they all state that theta is equal to curly brackets of some coordinate point. So, A is 0.3 pi divided by 4. B is 0.2 divided by 3, C is 65 divided by 6, and D is pi divided by 6.2 divided by 3. OK. So first off, in order to solve the equation, cosine of 2 theta minus cosine of theta is equal to 0, for the range of 0 is less than or equal to theta and theta is less than or equal to pi. We need to recall and use trigontric identities and some and algebra to help us perform the algebraic manipulation in order to solve for this particular problem. So with that in mind, we need to first recall the double angle identity for cosine. So, as we should recall, we should be able to note that cosine of 2 theta. is equal to 2 multiplied by cosine squared of theta is so cosine squared of theta minus 1. So, cosine of 2 theta is also equal to 2 multiplied by cosine square to theta minus 1. So now when we substitute this information into our equation, we will find that we could write that 2 multiplied by cosine squared of theta minus 1 minus cosine of theta. It's going to be all equal to 0. So now we need to rearrange our terms a little bit, and when we do, we will find that 2 multiplied by cosine squared of theta minus cosine of theta minus 1 is equal to 0. And as we should be able to recognize, this is a quadratic equation in terms of cosine of theta. So we need to state that X, so let's make a note that X is going to be equal to cosine of theta. So if we set X equal to cosine of theta, we should be able to write that 2 x 2 minus X minus 1 is equal to 0. Fantastic. So now we need to solve this quadratic equation using the quadratic function, and when we do that, we will simply find that, and once again, using the quadratic function to solve for X, we will find that X is equal to Once again, since you should be able to recall the quadratic function, so in case you can't recall it off the top of your head, the quadratic function states that X is equal to negative B plus or minus square root, B2 minus 4AC. All divided by 2 A. So using the quadratic function to isolate and solve for X, we will find that X is going to be equal to 1 plus or minus 3 divided by 4, which will mean that X is going to be equal to 1 and X is going to be equal to -12. So now we need to try both of these different values of X. So when we do that, when we when we plug in a value of X equals 1, we'll find that cosine of theta is equal to 1, so that will mean that theta is equal to 0, and for when X is equal to -12, that will mean that cosine of theta is equal to -12. Which will mean when we isolate and solve for theta, that will mean that theta is going to be equal to Inverse cosine of -12. And if we take this value, and it looks like we're running out of room, so let's move it over just a little bit here, so we could see it a lot better. OK. So that will mean that when we perform that will when we perform the equate like when we perform this expression, so essentially we're trying to find that theta is equal to, since we're trying to rearrange cosine theta equals -12 to isolate and solve for theta. So when we do that, we'll find that theta, the angle is equal to inverse cosine of -12. So that means the angle theta for which cosine theta equals -12 within the given interval will have to be, let's make a note of this in blue, that will mean that theta is going to have to be equal to 2 pi divided by 3. So therefore, the solutions for theta within the interval, 0 is less than or equal to theta and theta is less than or equal to pi are going to be. And let's make a note of this up above here. That will mean that. Let's make a note In green, so that will mean that theta has to be equal to 0 and theta has to be equal to 2 pi divided by 3. So that will mean, without a doubt, our final answer has to be multiple choice answer, letter B. Which states that theta is equal to curly brackets of 0.2 pi divided by 3. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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

cos 2θ + cos θ = 0

Key Concepts

Trigonometric Identities

Solving Trigonometric Equations

Unit Circle

Watch next