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CCSS.Math:

cosine of two theta is equal to C and theta is between zero and pi write a formula for sine of theta in terms of C and so I encourage you to pause the video and try to figure this out on your own before I work through it so I'm assuming you've had a go so let's see if we can work through it together and I'm going to get my scratch pad out I have that copy and pasted the exact question right over here and so let's think about it a little bit they're telling us that cosine of two theta is equal to C so let me write it this way C is equal to cosine of two theta now using my knowledge of angle addition formulas we know we know for example that cosine of alpha plus beta we know that cosine of alpha plus beta is equal to is equal to cosine alpha is equal to cosine alpha cosine beta cosine beta minus minus sine alpha sine beta minus sine alpha sine alpha sine beta sine beta now why would this be useful here well this is the sum of just theta plus theta and so I can rewrite this at least in terms of cosines and sines and maybe I can rewrite the cosines in terms of sines and then solve for the sines so let's try to do that so I can rewrite cosine of two theta that's the same thing as cosine of beta theta plus theta cosine of theta plus theta is of course the same thing as two theta and now I can use the angle addition formula for cosine this is going to be equal to this is going to be equal to cosine of theta cosine of theta times cosine of theta times cosine of theta minus sine of theta minus sine of theta times sine of theta times sine of theta which is of course equal to this is equal to cosine squared theta that's equal to cosine squared theta minus and what we have right over here is sine squared theta sine squared theta so let's see we've been able to rewrite C in terms of cosine squared theta and sine squared theta vote ideally we you just want to write it in terms of sine theta so that we can solve for sine theta so if we can re-express cosine theta in terms of sine well we already know from the Pythagorean identity that cosine squared theta plus sine squared theta is equal to 1 or we could say that cosine squared theta is equal to I'm just going to subtract sine squared from both sides is equal to 1 minus sine squared theta so let me rewrite this as 1 minus sine squared theta and then of course we have minus this yellow sine squared theta sine squared theta and all of this is equal to C all this is equal to C or we could get that C is equal to 1 minus 2 sine squared theta minus 2 sine squared theta and what's useful about this is now we just have to solve for sine of theta so let's see I could multiply both sides by a negative just so that I can switch the order over here so I could write this as negative C is equal to 2 sine squared theta 2 sine squared theta minus 1 it's multiplied both sides by negative and then let's see I could add 1 to both sides if I add 1 to both sides and I'll do I'll go over here if I add 1 to both sides I get 1 minus C 1 minus C is equal to is equal to 2 sine squared theta 2 sine squared theta I can divide both sides by 2 divide both sides by 2 and then so I get sine squared theta is equal to 1 minus C over 2 or I could write that sine of theta is equal to the plus or minus square root of 1 minus C over 2 of 1 minus C over 2 so that leads to a question is it both is it the plus and minus square root or is it just one of those and I encourage you to pause the video again in case you haven't already figured it out and look at the information here and and think about whether they give us the information of whether we should looking be looking at the positive or negative sign well they tell us they tell us that theta is between 0 and pi so if I were to draw a unit circle here between 0 and PI radians so at this angle right over here is 0 radians and pi is going all the way over here pi is going all the way over here so this this angle places it's terminal I guess it's terminal ray either in the first or second quadrants so it could be an angle like this it could be an angle like this it cannot be an angle like this and we know that the sine of an angle we know that the sine of an angle is the y-coordinate and so we know if we're in the first or second quadrant the y-coordinate is going to be non-negative so what we would want to take the positive square root right over here so we would get sine of theta sine of theta is equal to the principal root or you could even think of the positive square root of 1 minus C over 2 so let's go back to our make sure that we can let's check our answer so sine of theta is equal to the square root of 1 minus Capital whoops 1 minus capital C all of that over all of that over 2 and we got it right