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Amy has a master's degree in secondary education and has taught math at a public charter high school.
Now that you know a bit about trigonometry and its various functions, you might wonder about all the seemingly difficult trig problems that you might have encountered while flipping through other math books or looking at various math websites. Well, never fear! I will show you that some of those seemingly difficult problems are not so difficult after all.
In this video lesson, I will show you six sum and difference identities. These are trigonometric definitions that show how to find the sine, cosine, and tangent of two given angles that you can use to make your life easier.
Think of these identities as keys: if you didn't have the key, you would have to pick apart or calculate each and every part of the problem, but with the key, you have just one component to unlock or calculate. Are you ready to find out what these six keys are? Okay! Let me show you:
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The alpha and beta symbols represent angles. Look at the left side of the equations. Aren't these much simpler than the right side of the equations? The way these identities help you is if you see anything that looks like the right side, you can go ahead and substitute the left side into the problem, as most likely you will be able to solve the left side instead of the right side. How can you remember these identities?
Look for patterns in these identities that you can easily remember. For example, the plus and minus signs of the sine identities are the same on both sides of the equation, while the plus and minus signs are opposite for the cosine identities. The tangent has matching plus and minus signs in the numerator and opposite signs in the denominator. What other patterns do you see?
Now that we've covered our identities, we need now to think back to our unit circle, a circle with a radius of 1. With most trig problems that are this complicated, you will most likely be working with the unit circle in either degrees or radians. In my experience, the radian measure for angles is more commonly seen. To help you refresh your memory of the unit circle, here are two unit circles: one for degrees and one for radians:
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Remember that to use these unit circles, you look for your angle inside the circle, and then you refer to the point on the circle for your answer. The points give you the answer for the cosine and sine functions for that angle. The x part is the cosine of the angle, and the y part is the sine of the angle.
For example, looking at the angle 5pi over 3, we see that our point is (one half, negative square root of 3 divided by 2). So, this means that for this angle, the cosine equals one half and the sine equals minus the square root of 3 over 2. Also, remember that since tangent equals sine divided by cosine, you can also easily use the unit circle to find your tangent values.
Let's look at how we can use our identities and our unit circle to answer some seemingly difficult problems. First, we have this problem:
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At first glance, this problem looks complicated. You look at your angles; they are written in radian measure. You look at your unit circle, and you don't see pi over 12 anywhere. You could go ahead and punch all of this into your calculator, but you would get a whole string of numbers.
What else can you do to get a more exact answer? Ah, your identities! Yes, you stare at this for a while, and then you realize that this is the sum identity for the sine function. Your alpha angle is pi over 12, and your beta angle is also pi over 12.
The identity tells you that this whole string simply equals the sine of the alpha angle plus the beta angle. So, we have the sine of pi over 12 plus pi over 12. This equals the sine of 2pi over 12, which simplifies to the sine of pi over 6.
Now, you look at your unit circle again and lo and behold! There is your pi over 6 angle. Your sine is the y part of the point. What does it equal? It equals one half. Our answer is one half. Yay!
Let's look at another one:
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This one looks complicated, as well, at first glance. You see the angles listed on the unit circle, but is there an easier way for you to solve this problem? Yes, there is! You go to your identities again. Aha! This identity is for the sum of cosines.
This whole string is equal to the cosine of the alpha angle plus the beta angle. The alpha angle equals pi over 6, and the beta angle equals pi over 3. The alpha angle plus the beta angle equals pi over 2, then.
So, all we need to calculate to find our answer is cosine of pi over 2. We look at our unit circle and we find our pi over 2 angle. What is our cosine at this point? It is 0. Our answer is simply 0. Very nice!
Let's review what we've learned now. We learned that our sum and difference identities are trigonometric definitions that show how to find the sine, cosine, and tangent of two given angles. There are six of them in total: two for sine, two for cosine, and two for tangent. We remember them by looking at their patterns. Here is the list of them again:
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We use these in conjunction with the unit circle, our circle of radius 1. On our unit circle, we have labeled certain angles that have easy answers. What usually happens with these identities is that once you add or subtract the alpha and beta angles, you will get an angle that is on the unit circle. Once you are at this point, you can easily find your answer by referring to the unit circle.
After you have finished with this lesson, you'll be able to:
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