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# Using the cosine angle addition identity

Sal evaluates the cosine of the sum of 60° and another angle whose right triangle is given. To do this, he must use the cosine angle addition formula. Created by Sal Khan.

## Want to join the conversation?

• At , when finding the cosine of 60 degrees, and at . when finding the sine of 60 degrees, why do you use the example triangle that you drew? Why not just use the triangle given with the problem?
• The triangle given in the problem doesn't have a 60 degree angle.
• Why can you not just say that the cosine of B is 15/17, take the inverse to get the angle, add 60 and take the cosine of all of that? You get the same answer. Will this not always work?
• It will, but this is showing how to get the exact answer not a repeating decimal.
• Why is this so important to learn for life?
• The field of robotics requires knowlage of kinkamatics in order for a programmer and/or mechanical enginear to effectively program machinery. It takes a great deal of math to tell an artificial hand exactly how to help a disabled veteran tie his shoe.

The artificial hand has to know where it's parts are in virtual space, as compared to real space. This requires a working knowlage on how to mathematically describe angles and their position in third dimentional space.
• At how did you determine that 8/17 multiplied by square root of 3 equaled 4 square root of 3 over 17?
• In multiplication you can divide the factors with each other. I can't really show you since I'm limited by a keyboard, but I'll try:
8/17* sqrt 3/2 <=>
8/2*sqrt 3/17 and 8/2 is 4, which he multiplied with sqrt 3/17. He's not doing anything fancy, just writing a fraction directly in simplest form. Excuse my bad English, I'm doing my best to explain.
• I've always wondered, why can't we just use the distributive property and simplify cos( x + y) to cos( x) + cos(y)?
• Because you can't distribute that function. It's similar to how (x + y)^2 is not equal to x^2 + y^2.
• Couldn't we just use the law of sines here? We have all the sides and one angle, so can't we find the other angles, add 60, and call it good that way?
• You can use the law of sines as well.

Here are the other ways to solve this problem I can think of:
- Using the law of cosines
- Using the inverse trig functions

One good thing of the way used in the video is that you don't have to use inverse functions, hence you don't get repeating decimal.
• Isn't it easier to just take arccos(15/17)≃28.07°, and then cos(28.07°+60°)?
• Yes, but the cosine angle identity could still help if you want an EXACT answer.
• if that's assumed to be a 30 60 90 then the cos(<abc + 60) = cos(90) then will the last simplified terms at simplify to 0?
(1 vote)
• Yes, that would be true if the triangle was a 30 60 90 triangle and <abc was the 30 degree angle. However, you can't assume that that's the case in this problem. The 8, 15, 17 side lengths don't align with the x, xsqrt(3), and 2x side lengths that every 30 60 90 triangle has, meaning that <abc is not 30 degrees in the problem.