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CCSS.Math:

let's say that I've got some angle theta some angle theta right over here and I'm drawing it on our unit circle with the typical convention that we started with array that's along the positive x-axis and the terminal side of this angle is the terminal side of the angle where it intersects the unit circle determines essentially the sine and cosine of that theta so the cosine of theta is the X is Z I'm just going to color I haven't used before the cosine of theta is the x-coordinate of where this terminal ray intersects the unit circle or another way of thinking about it is the cosine of theta is the length of what I'm drawing in purple right over here it's this length that length right over there is cosine of theta and the sine of theta is the y-coordinate or another way of thinking about it the sine of the sine of theta is the length of this line right over here the how high you are above the x-axis that is essentially the y-coordinate and so the length of that is sine theta and this makes sense this is actually this actually shows why the unit circle definition is an extension of the sohcahtoa definition remember sohcahtoa let me write it down sohcahtoa so huh so Toa tow tells sine is opposite over hypotenuse so if I want to do the sine of theta what's it going to be so if I think about the sine of theta sine of theta by the sohcahtoa definition it's going to be equal to the length of the opposite side well we're saying that that's sine of theta its sine of theta over the hypotenuse well the hypotenuse here this is a unit circle so it's going to be 1 so this shows that this is consistent or another way of thinking about it it is or another way of thinking about sine of theta is equal to the opposite side over the hypotenuse in this case it's going to be equal to the opposite side and what's the hypotenuse this is a unit circle so it's going to be 1 so in this case sine of theta is equal to the length of the opposite side the length of the that side is equal to sine theta and same exact logic the cosine of theta the cosine of theta is equal to adjacent over hypotenuse is equal to adjacent over hypotenuse and so that's since the hypotenuse is equal to one that's just the length of the adjacent side so cosine of theta is the length of the adjacent side so this is all a little bit of review just showing how the unit circle definition is an extension of the sohcahtoa definition but now let's do something interesting this is the angle theta let's think about the angle theta plus pi over two so the angle theta plus PI over two so if I were to essentially add PI over two to this I'm going to get a ray that is perpendicular to the first ray PI over two if we think in degrees PI over two radians so when I say theta plus PI over two I'm talking in radians PI over two radians is equivalent to ninety degrees so we're essentially adding 90 degrees to it so this angle right over here that angle right over here is theta plus PI over two now what I want to explore in this video and I guess this is the interesting part of the video is can we relate sine of theta plus PI over two to somehow sine of theta or cosine of theta and I encourage you to pause this video and try to think this through on your own before I work work it out well let's think about what sine of theta plus PI over two is we know from the unit circle definition the sine of this angle which is theta plus PI over two is the y-coordinate it's that it's this value right over here or another way of thinking about it it's the length it's the length of this line in magenta this right over here is the sine of theta plus PI over 2 so that right over there now how does that relate to what we have over here well when you look at it looks like we just took this triangle and we just kind of we rotated it we rotated it counterclockwise by 90 degrees which is essentially what we did do because we took this terminal side and we added nine agrees to it or PI over two radians and if you want to get a little bit more rigorous about it if this whole if this whole white angle here is theta plus PI over two and the part that's in the first quadrant is PI over two then this part right over here that must be equal to theta and if we think about it if we try to relate this side this side that I put in magenta relative to this angle theta using the sohcahtoa definition here relative to this angle theta in yellow this is the adjacent side so let's think about it a little bit so if we were so what deals with the adjacent and the hypotenuse and in this case of course our hypotenuse has length one this is a unit circle well cosine deals with adjacent and hypotenuse so we could say that the cosine of this theta so the cosine of that theta is equal to the adjacent side the length of the adjacent side which we already know is sine of theta plus PI over two right this way sine sine of theta plus PI over two over the hypotenuse over the hypotenuse which is just one so that doesn't change its value so that was pretty neat just like that we were able to come up with a pretty neat relationship between cosine and sine the cosine of theta is equal to sine of theta plus PI over two or you could say sine of theta plus PI over two is equal to cosine of theta now what I encourage you to do is after this video see if you can come up with other other results think about what happens to how what what sine of theta really relates to or or what cosine of theta plus PI over two might relate to so I encourage you to explore that on your own