Alternate Forms of Trigonometric Identities

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  • 0:54 Alternate Forms
  • 2:32 Half Angle Indentities
  • 3:25 Using Alternate Forms
  • 4:37 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will know some of the alternate forms of the basic trigonometric identities. You will also learn how to use these alternate forms.

Trigonometric Identities

Think of some of the trigonometric identities, or true statements used in trigonometry, that you've learned and you might recall that they are a challenge to remember. Some of the trigonometric identities involve quite a few components, don't they? For example, just look at this half angle identity:

alternate trig identities

But once you remember them, they actually serve you quite well. They help you solve more complicated trigonometric problems. You can make substitutions that will simplify your equations, which will help you break down the problems into simpler ones that you can solve.

To help you even more, here are some alternate forms of our trig identities. Sometimes you need these alternate forms to help you solve problems. After all, problems come in all shapes and sizes. They can be written in alternate ways that require you to know these alternate forms so you can make the appropriate substitutions.

Alternate Forms of Trig Definitions

Are you ready for more things to go inside your brain? Okay, let's begin.

We'll start with alternate ways of expressing our trig definitions. Recall that all of trigonometry begins with just two functions, the sine and cosine. The rest can be found from these two functions. We have our tangent function equal to sine over cosine. Then we also have cosecant equal to 1 over sine. Secant is 1 over cosine, and cotangent is 1 over tangent. These are our basic trig definitions:

alternate trig identities

By making substitutions to these basic definitions and rearranging, we come up with the alternate forms of our definitions. We have our sine equal to 1 over cosecant. Cosine is equal to 1 over secant. And tangent is equal to 1 over cotangent. What we've done here is simply reverse our beginning definitions for cosecant, secant, and cotangent:

alternate trig identities

One other alternate form is that for the cotangent. Because it is the reciprocal of the tangent, we can also define it as the reciprocal of the tangent definition, or cosine divided by sine.

alternate trig identities

And there we have our alternate forms of our trig definitions. How can you remember these? Think of flipping the definitions around. You know that cosecant is 1 over the sine function. So now think of solving this for the sine function to see what you get. Do this for each of the basic definitions, and you will get our alternate forms.

Alternate Forms of Half Angle Identities

Let's look at one last set of alternate forms. These alternate forms are for the half-angle identities. They are called half-angle identities because the argument inside our function is a half angle.

alternate trig identities

How can we write these differently? Well, look at the square root; we don't like square roots. How can we get rid of the square root? We can eliminate the square root by squaring both sides of our equation. We can also eliminate the fraction by multiplying all the angles by 2. When we do that, we get these alternate forms:

alternate trig identities

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