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# Finding trig values using angle addition identities

CCSS.Math:

## Video transcript

what I want to attempt to do in this video is figure out what the sine of 7 PI over 12 is without using a calculator and so let's just visualize 7 PI over 12 on the unit circle so one side of the angle is going along the positive x-axis and then let's see if we go straight up that's PI over 2 which is the same thing as 6 PI over 12 so then we essentially just have another PI over 12 to get right over there this is the angle that we're talking about that is 7 PI over 12 radians and the sign of it by the unit circle definition of sine it comes it's the y coordinate of where this ray intersects the unit circle so is the unit circle has radius one where it intersects the unit the y coordinate is the sine so another way to think about it it's the length it's the length of this line right over here and I encourage you to pause the video right now and try to think about it on your own see if you can use your powers of trigonometry to figure out what sine of 7 PI over 12 is or essentially the length of this magenta line so I'm assuming you've given a go at it and if you're like me your first temptation might have been just to focus on this triangle right over here that I kind of drew for you so the triangle looks like this it looks like this where that's what you're trying to figure out this length right over here set sine of 7 PI over 12 we know the length of the we know the length of the hypotenuse is 1 it's a radius of the unit circle it's a right triangle right over there and we also know this angle right over here which is this angle right over here this gets a 6 PI over 12 then we have another PI over 12 so we know that that is PI over 12 not PI over 16 we know that this angle right over here is PI over 12 and so given this information we can figure out this this or we can at least relate this side to these other sides using a trig function relative to this angle this is the adjacent side and so the cosine of PI over 12 is going to be this magenta side over 1 or you could just say it's equal to this magenta side so you could say this is cosine of pi over 12 so we just figured out that sine of 7 PI over 12 is the same thing as cosine of PI over 12 but that still doesn't help me I don't know offhand what the cosine of PI over 12 radians is without using a calculator so let's instead of thinking about it this way let's see if we can compose this angle or if we can decompose it into some angles for which we do know the sine and cosine and what angles are those well those are the angles and special right triangles so for example we are very familiar with 30 60 90 triangles 30 60 90 triangles look something like this this is my best attempt at hand drawing it if they have so instead of writing the 30-degree side I'll write since we're thinking in radians I'll write that as PI over 6 radians the 60-degree side I'm going to write that as PI over 3 radians and of course this is the right angle and if the hypotenuse here is 1 then the side opposite the 30-degree side of the PI over 6 radians side is going to be 1/2 the hypotenuse which in this case is 1/2 and then the other side that's opposite the 60-degree side or the PI over 3 radians side is going to be square root of 3 times the shorter side so it's going to be square root of 3 over 2 and so we've used these types of triangles in the past to figure out the sine or cosine of 30 or 60 or in this case PI over 6 or PI over 3 so this we know about PI over 6 and PI over 3 we also know about 45-45-90 triangles we know that they're isosceles right triangles they look like this my best attempt at drawing it that one actually doesn't look that isosceles so let me let me make it let me make it a little bit more so I don't know that looks closer to being ice and isosceles right triangle and we know if the length of the hypotenuse is 1 and this comes straight out of Pythagorean theorem then the length of each of the other two sides are going to be square root of 2 over 2 times the hypotenuse which is this case is the square root of 2 over 2 instead of describing these as 45-degree angles we know that's the same thing as PI over 4 PI over 4 radians and so if you give me PI over 6 PI over 3 PI over 4 I can use these triangles either using the classic definition sohcahtoa definitions or I could stick them on the unit circle here to use the unit circle definition of trig functions to figure out what the sine cosine or tangent of these angles are so can I decompose 7 PI over 12 into some combination of pi over 6 is PI over 3 s or PI over 4 s well to think about that let me rewrite PI over 6 PI over 3 and PI over 4 with the denominator over 12 so let me write that so PI over 6 is equal to 2 PI over 12 PI over 3 is equal to 4 PI over 12 and PI over 4 is equal to 3 PI over 12 so let's see 2 plus 4 is not 7 2 plus 3 is not 7 but 4 plus 3 is 7 so I could use this and this 4 PI over 12 plus 3 PI over 12 is 7 PI over 12 so I could rewrite this this is the same thing as sine of 3 PI over 12 3 PI over 12 plus 4 PI plus 4 PI over 12 which of course is the same thing sine of PI over 4 I'll do this in another color sine of PI over 4 plus plus let me do this plus PI over 3 plus PI over 3 and now we can use our angle addition formula for sine in order to write this as the sum of products of cosines and sines of these angles so let's actually do that so this right over here is going to be equal to this is going to be equal to the sine the sine of PI over 4 times the cosine of PI over 3 times the cosine of PI over 3 plus the other way around cosine of PI over 4 cosine of PI over 4 times the sine of PI over 3 sine of PI over 3 so now we just have to figure out these things and I've already set up the triangles to do it what is sine of PI over 4 sine of PI over 4 well let's let's think about this is PI over 4 right over here sine is opposite over hypotenuse well it's just going to be square root of 2 over 2 so this is square root of 2 over 2 square root of 2 over 2 what is cosine of PI over 3 well this is a PI over 3 radians over here cosine is adjacent over hypotenuse so this is going to be its adjacent over hypotenuse so this is going to be one half what is cosine of PI over 4 we'll go back to PI over 4 so it's adjacent over hypotenuse it's square root of 2 over 2 it is also square root of 2 over 2 square root of 2 over 2 and what's sine of PI over 3 well sine sine is opposite over hypotenuse so it's square root of 3 over 2 over 1 square root of 3 over 2 divided by 1 which is square root of 3 over 2 and so now we just have to simplify all of this business so this is going to be equal to this is going to be equal to the sum of this or the product I should say is just square root of 2 over 4 and then plus plus the product of these let's see we could rewrite that is we could write that as square root of 6 over 4 square root of 6 over 4 or we could just rewrite this whole thing as and we deserve our we deserve a little bit of a we deserve a little bit of a drumroll at this point this is equivalent to let me scroll over to the right a little bit this is equivalent to square root of 2 plus square root of 6 plus square root of 6 all of that all of that over 4 that's what sine of 7 PI over 12 is or cosine of PI over 12 what that is equal to