Factor each trigonometric expression.
4 tan² β + tan β -...

Question

Factor each trigonometric expression.

4 tan² β + tan β - 3

Accepted Answer

Welcome back. I am so glad you're here. We're asked to rewrite the following trigonometric expression by factoring it completely. Our given expression is five cotangent squared X minus cotangent X minus four. Our answer choices are answer choice. A, the quantity of five cotangent X plus four multiplied by the quantity of cotangent X minus one answer. Choice B, the quantity of five cotangent X minus four multiplied by the quantity of cotangent X plus one answer choice C, the quantity of five cotangent X plus four multiplied by the quantity of cotangent X plus one. And answer choice D, the quantity of cotangent X plus four multiplied by the quantity of four cotangent X minus one. All right. So we've got a trinomial here and we need to factor it and we've got these cotangents and to make it a little bit easier to look at, we can use the method of use substitution, we can substitute this middle term cotangent X set that equal to U and we'll use that for a moment. So this is a little bit easier to conceptualize. So instead of having five multiplied by cotangent squared X, since cotangent of X is U, that'll be U squared. So we have five U squared minus instead of cotangent of X, that'll be minus U and then minus four. And then we want to have two open sets of parentheses where we can factor this, we're doing the reverse of the foil process. So we ask ourselves what multiplied by what gives us this first term five U squared. And that's going to be a five U multiplied by A U. And then we're going to look at our lasts and ask ourselves what multiplied by what gives us a negative four. But when we take our outsides plus our insides adds up to a negative one U. And so if you can play around with this a little bit, if you need to definitely check out previous lesson videos on factoring if you need a refresher. But what this is going to be is we're going to have the positive four multiplied by a negative one for our lasts. And that's going to give us that minus one U. And you can always just foil this out really quick to check our first five U multiplied by U is five U squared outsides five U multiplied by negative one is minus five U insides four U multiplied by one is plus four U and then our lasts four, multiplied by negative one is negative four. Simplifying here we bring down our five U squared negative five U plus four U is going to be minus one U and then bring down our minus four. So that checks out, we factored it correctly. Now, all we need to do is go back and substitute in for our use, what U is equal to which is the cotangent of X. So plugging that cotangent X back in, we have five, not multiplied by U but by cotangent X plus four, all in one set of parentheses multiplied by the quantity of, instead of U, we have cotangent X minus one. And there we have our trigonometric expression fully factored. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Factor each trigonometric expression.
4 tan² β + tan β - 3

Key Concepts

Trigonometric Functions

Factoring Quadratic Expressions

Zero Product Property

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