How to Solve Trigonometric Equations by Factoring

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One way to effectively solve a trigonometric equation is through the careful use of factoring. Solve trigonometric equations by factoring with help from a mathematics educator in this free video clip.

Part of the Video Series: Trigonometry Education
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Hi, I'm Jimmy Chang, and we're here to talk about how to solve trigonometric equations by factoring. Now, when it comes to factoring using trigonometric equations, you actually factor the same way you would regular algebraic non-trigonometric equations, even though it might take a little getting used to. But what you want to think about is using the things you've learned about trigonometric values and identities, and you might need to take the inverse trig functions to find out the actual angles that you're looking for. Here's an illustration: Suppose you have the equation, sine squared theta minus sine theta minus two is equal to zero. Now, you actually solve this and factor it the same way that you would back in the old algebra days, because sine square theta, for example, breaks up into sine theta and sine theta. And, what you want to think about is, factors of minus two, that's going to add to give you negative one, because there's actually a one in here. So, what you want to think about is the only two numbers that will fit is, minus two plus one. And once you have that, since both of them are set equal to zero, you can take each equation and set equal to zero, and then try to solve for theta, if possible. Now, in this particular case in both equations, you want to get the sine theta on its own. So, in this case, you want to add two to both sides. So, you have sine theta is equal to two. Now, this should give you a warning sign, because in all the times you've studied trigonometry, there's no sine value of an angle that's going to give you two. So, this actually has no solution. Now, for this one, we're going to hope for a better answer. Subtract one on both sides. So, you have sine of theta is equal to a negative one. Now, at this particular point, you want to think about sine of what angle is going to give you negative one? The only angle that's going to work there is three pi over two, because you know sine of three pi over two is equal to negative one. So as a result, the angle here the answer is, theta is equal to three pi over two. And that's the only solution we have in this particular equation. So, the bottom line is, when you're trying to solve trigonometric equations by factoring, you factor using the normal methods that you might have learned before, and then solve for theta or the angle, whenever possible. So, I'm Jimm Chang, and that's a brief overview as to how to solve trigonometric equations by factoring.

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