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Studying for a test? Prepare with these 12 lessons on Right triangles & trigonometry.
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We are told that the cosine of 58 degrees is roughly equal to 0.53. And that's roughly equal to, because it just keeps going on and on. I just rounded it to the nearest hundredth. And then we're asked, what is the sine of 32 degrees? And I encourage you to pause this video and try it on your own. And a hint is to look at this right triangle. One of the angles is already labeled 32 degrees. Figure out what all of the angles are, and then use the fundamental definitions, your sohcahtoa definitions, to see if you can figure out what sine of 32 degrees is. So I'm assuming you've given a go at it. Let's work it through now. So we know that the sum of the angles of a triangle add up to 180. Now in a right angle, one of the angles is 90 degrees. So that means that the other two must add up to 90. These two add up to 90 plus another 90 is going to be 180 degrees. Or another way to think about is that the other two non-right angles are going to be complementary. So what plus 32 is equal to 90? Well, 90 minus 32 is 58. So this right over here is going to be 58 degrees. Well, why is that interesting? Well, we already know what the cosine of 58 degrees is equal to. But let's think about it in terms of ratios of the lengths of sides of this right triangle. Let's just write down sohcahtoa. Soh, sine, is opposite over hypotenuse. Cah, cosine, is adjacent over hypotenuse. Toa, tangent, is opposite over adjacent. So we could write down the cosine of 58 degrees, which we already know. If we think about it in terms of these fundamental ratios, cosine is adjacent over hypotenuse. This is a 58 degree angle. The side that is adjacent to it is-- let me do it in this color-- is side BC right over here. It's one of the sides of the angle, the side of the angle that is not the hypotenuse. The other side, this over here, is a hypotenuse. So this is going to be the adjacent, the length of the adjacent side, BC, over the length of the hypotenuse. The length of the hypotenuse, well, that is AB. Now let's think about what the sine of 32 degrees would be. Well, sine is opposite over hypotenuse. So now we're looking at this 32 degree angle. What side is opposite it? Well, it opens up onto BC. And what's the length of the hypotenuse? It's AB. Notice, the sine of 32 degrees is BC over AB. The cosine of 58 degrees is BC over AB. Or another way of thinking about it, the sine of this angle is the same thing as the cosine of this angle. So we could literally write the sine-- I want to do that in that pink color-- the sine of 32 degrees is equal to the cosine of 58 degrees, which is roughly equal to 0.53. And this is a really, really useful property. The sine of an angle is equal to the cosine of its complement. So we could write this in general terms. We could write that the sine of some angle is equal to the cosine of its complement, is equal to the cosine of 90 minus theta. Think about it. I could change this entire problem. Instead of making this the sine of 32 degrees, I could make this the sine of 25 degrees. And if someone gave you the cosine of-- what's 90 minus 25?-- if someone gave you the cosine of 65 degrees, then you could think about this as 25. The complement is going to be right over here. This would be 65 degrees. And then you could use the exact same idea.