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# Using complementary angles

CCSS.Math:

## Video transcript

we are told that the cosine of 58 degrees is roughly equal to zero point five three and that's roughly equal to because it just keeps going on and on I just rounded it to the nearest hundredth and then there 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 could figure out what sine of 32 degrees is so I'm assuming you've given a go at it let's work on 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 99 these two add up to 90 plus another 90 is going to be 180 degrees or another way to think about it 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 so sine is opposite over hypotenuse 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 the 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 the hypotenuse so this is going to be the adjacent the length of the adjacent side BC over the length of the hypotenuse over the length of the hypotenuse the length of the hypotenuse well that is a B now let's think about what the sine of 32 degrees would be so the sine of 32 degrees well sine is opposite over hypotenuse so now we're looking at this 32 degree angle what side is opposite well it opens up onto BC it opens up onto BC just like that and what's the hypotenuse well we've already or the length of the hypotenuse it's a B it's a B notice the sine of 32 degrees is BC over a B the cosine of 58 degrees is BC over a B or another way of thinking about 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 the sine of 32 degrees is equal to the cosine cosine of 58 degrees which is roughly equal to 0.5 3 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 have done I could change 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