Verifying Trigonometric Identities - Double Angle Formulas

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來自 University of California, Irvine 的課程

Pre-Calculus: Trigonometry

257 個評分

University of California, Irvine

Pre-Calculus: Trigonometry

257 個評分

This course covers mathematical topics in trigonometry. Trigonometry is the study of triangle angles and lengths, but trigonometric functions have far reaching applications beyond simple studies of triangles. This course is designed to help prepare students to enroll for a first semester course in single variable calculus. Upon completing this course, you will be able to: 1. Evaluate trigonometric functions using the unit circle and right triangle approaches 2. Solve trigonometric equations 3. Verify trigonometric identities 4. Prove and use basic trigonometric identities. 5. Manipulate trigonometric expressions using standard identities 6. Solve right triangles 7. Apply the Law of Sines and the Law of Cosines

從本節課中

Basic Trigonometric Identities II

In this module, we continue our exploration of trigonometric function identities. We will begin by learning how to verify such identities. We will then talk about the double-angle and half-angle identities.

Verifying Trigonometric Identities- Sum & Difference 3:55

Verifying Trigonometric Identities - Double Angle Formulas 7:20

Half Angle Identities, Part 18:25

Half Angle Identities, Part 2 5:23

與講師見面

Dr. Sarah Eichhorn
Associate Dean, Distance Learning & Tenured Lecturer
Dr. Rachel Cohen Lehman

Let's look at verifying trigonometric identities that involve the Double Angle Formulas.

For example, let's verify this identity.

Well, we can start working with the left hand side here.

This is equal to sine x plus cosine x,

times sine x plus cosine x.

And now we can foil,

so this is sine squared x plus sine x cosine x,

and then plus cosine x sine x and then plus cosine squared.

Now, sine x cosine x and cosine x sine x are the same.

So, we'll have two of them.

That is, this is equal to sine squared x plus

2 sine x cosine x

and then plus cosine squared x.

Now, let's group these two together.

In other words, this is equal to sine squared x plus cosine

squared x and then plus 2 sine x cosine x.

Now, why did we do that?

Because remember, we have the following Pythagorean Identity,

sine squared x plus cosine squared x is 1,

which we can use here,

as this is just one.

So, this is equal to 1

plus 2 sine x times cosine x.

Now, remember the Double Angle Formula

for sine namely that

sine 2x is equal to 2 sine x cosine x.

So, let's use that here.

Which means that this is equal to 1 plus sine 2x which is our right hand side.

And therefore, we verified this identity.

Okay, let's see another one.

Let's verify this identity.

We're going to start working here on the right hand side.

Now, looking in the denominator here,

there is another Pythagorean Identity that will help us,

namely 1 plus tangent

squared x is equal to secant squared x.

So, let's use that here.

Namely, this is equal to 1 minus tangent

squared x divided by secant squared x.

And now, let's divide both terms in the numerator by secant squared x,

which gives us 1 divided by secant

squared x minus tangent squared x divided by secant squared x.

And now, let's convert everything to sine and cosine.

In other words, this is equal to 1 over secant x is cosine x, and so,

1 over secant squared x is cosine squared x, and then minus,

now tangent x is sine x over cosine x,

so tangent squared x is sine squared x over cosine squared x,

and secant squared x is 1 over cosine squared x.

Now, what is this simplified to?

Remember, when we divide fractions,

it's the same as multiplying the numerator by the reciprocal of the denominator.

In other words, this is sine squared x divided by cosine squared x,

and then times cosine squared x over 1.

Now, won't these cosine squared x's cancel?

So, all we're left with then is just sine squared x.

That is this is equal to cosine squared x minus sine squared x.

Now, let's recall all the Double Angle Formulas.

Unlike sine and tangent,

cosine has three different double angle formulas.

But if we look at them,

what we just got is equal to the right hand side of the first one,

which means that what we just got is equal to cosine of 2x.

So, this is equal to

cosine 2x which is our left hand side.

And so, we verified this identity.

And this is how we verify a trig identities that involve the Double Angle Formulas.

Thank you, and we'll see you next time.

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