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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

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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.

Watch this video lesson, and you will see how the angle sum and difference identities are derived for sine, cosine, and tangent. You will also learn how you can use these identities to help you solve problems.

Trigonometry is full of identities or definitions. The ones that you are learning about right now have been proved over time and shown to be true. Because these identities have been mathematically derived with formal proofs, you can use these identities without worrying that they may give you a wrong answer.

The set of identities that we will look at today is called the **angle sum** and **difference identities**. These identities define how to turn a trig function of two angles added or subtracted from each other into a trig function of single angles. There are a total of six of these identities. Two for each of our trigonometric functions of sine, cosine, and tangent. There are two because we have one for addition and one for subtraction. Let's take a look at what they look like.

The alpha and beta symbols represent angles written in either radians or degrees. Do your best to memorize these. It will help you in the long run on tests and exams. The key to memorizing these is to look for patterns.

For example, the cosine sum and difference identities splits into a pair of cosines and then a pair of sines. If we have the two angles added together, then our pair of sines will be subtracted from our pair of cosines. So, the sign flips for cosine. Looking at our sine identities we see that the sign stays. If we have the two angles being added, then the definition part also has a plus. What other patterns do you see?

Remember how we talked about how these identities have been proven or derived mathematically so you can use these with confidence? Well, here is an overview of a simple way to prove these angle sum and difference identities. I don't expect you to fully understand these proofs as these derivations take you into the realm of higher math such as complex exponentials and imaginary numbers. But, if you can roughly see how these identities have been proven, then it will give you that much more assurance that these identities will work for you and will not fail you.

Let's take a look at the proof. It only requires four lines. It makes use of complex exponentials. It also makes use of the Euler formula, which states that *e^ix = cos x + i sin x*. Don't worry if these don't make sense now. Once you delve into higher math, such as calculus and higher, then you will understand how this all ties in with what you are learning now. Okay, so here is the proof:

Study this proof carefully and we can see that yes, it uses the higher math skills, but it also uses our basic algebra skills to multiply things out. You might be asking how does this get to our identities? Well, take a look at the last line and see how we have two sets of parentheses. Now look at the very first line, the left side of the equation gives us our cosine sum and sine sum. If we set the cosine sum equal to the first set of parentheses in the last line, we see that we get our cosine sum identity. If we set the sine sum equal to the second set of parentheses in the fourth line, we see that we get the sine sum identity.

To get our difference identities, we just replace our beta angle with a minus beta angle. To get our tangent identities, we make use of the definition of tangent in terms of sine and cosine. We know that tangent equals sine/cosine, so we simply write our tangent sum identity as the sine sum identity over the cosine sum identity. We then simplify to get to our tangent identity.

Pretty neat, isn't it?

Now you know that these identities are valid; what can you do with them? These identities are very useful for helping you to solve trig problems. Of course, with the help of calculators, you can solve any kind of problem by punching it in. But see, back in the day, people didn't have the use of handy graphing calculators that can do all kinds of fancy calculations. They had to work things out by hand. So, these identities helped them do that.

Where they couldn't evaluate the trig function of the sum of a pair of angles, they could evaluate these angles separately, and vice versa. Where they couldn't evaluate the trig functions for the single angles, they could evaluate the trig function for the sum of the angles. On some tests and exams that you will take, the same applies. You may not be able to use a calculator, so you will need to rely on these identities to help you solve trig problems. You want to see how this works?

Let's take a look at an example.

Looking at this problem, we see that this is the right side of the cosine difference identity. We also see that the angles by themselves are not easy to evaluate without a calculator. So what do we do? We use our cosine difference identity to help us. By using that identity, we find that our whole phrase equals the *cosine of 5 pi over 12 minus pi over 12*. This simplifies to the *cosine of 4 pi over 12*. This simplifies even further to the *cosine of pi over 3*. This we can evaluate with the aid of our unit circle, our special circle with a radius of 1 that includes the angles that have clean answers for cosine and sine. We see that the cosine of pi over 3 equals 1/2. And we are done!

That was pretty cool, wasn't it? We went from a problem that seemed very difficult to a very easy problem to solve. And remember, if you see pi in your angle, then you are dealing in radians. If you are using a calculator, make sure your calculator is set to calculate in radians and not degrees.

Let's review what we've learned.

We learned that the **angle sum and difference identities** define how to turn a trig function of two angles added or subtracted from each other into a trig function of single angles. We have a total of six of them.

The proof of these identities involves the use of complex exponentials as well as the use of the Euler formula.

With the help of these identities, we can take problems that we can't evaluate easily and turn them into something that we can. We can either have a problem with single angles that we can't evaluate by themselves, but we can evaluate it if we added or subtracted the angles. And vice versa - we might get a problem where we can't evaluate the two angles added or subtracted, but we can evaluate the angles by themselves. Without using a calculator, we can solve our problems with the help of our unit circle, the special circle of radius 1.

After this lesson is over you should be able to:

- State the angle sum and difference identities for sine, cosine, and tangent
- Recall the validity of the proof for the angle sum and difference identities
- Use the angle sum and difference identities to solve triangle problems

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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

- Solving Oblique Triangles Using the Law of Cosines 6:48
- Solving Real World Problems Using the Law of Cosines 6:37
- Using the Law of Sines to Solve a Triangle 7:02
- Solving Real World Problems Using the Law of Sines 7:09
- Proving the Addition & Subtraction Formulas for Sine, Cosine & Tangent 6:35
- Go to Trigonometric Applications

- Go to Continuity

- Go to Limits

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