now starting with the power reducing formula of sine squared we're going to get the half angle formula so sine squared we know it's 1 minus cosine 2 theta divided by two so that's true then sine squared theta divided by two must be equal to one minus cosine theta over two so if we divide this angle by 2 then we should divide that angle by 2. now all we need to do at this point is take the square root of both sides so now we have the half angle for sine sine theta divided by 2 is plus or minus the square root of 1 minus cosine theta divided by 2. so that's one of the formulas that you need to know that's the half angle formula for a sine now let's do the same with cosine let's start with the power reducing formula of cosine squared so it's 1 plus cosine 2 theta divided by 2. so let's divide both angles by two so this becomes cosine theta over two and two theta will change into theta now let's take the square root of both sides so cosine theta divided by two is plus or minus the square root of one plus cosine theta divided by two so that's the half angle of cosines tangent theta divided by 2 is equal to sine theta over 2 divided by cosine theta over 2. and sine is plus or minus the square root of one minus cosine theta over two and the formula for cosine is plus or minus the square root one plus cosine theta over two so we can get rid of the twos so what we have now is the square root of one minus cosine divided by the square root of 1 plus cosine and all we need is a single plus or minus sign in front now let's say if we have the square root of 5 divided by two this is the same as the square root of five over two so therefore we could say that the half angle formula for tangent is plus or minus square root one minus cosine divided by 1 plus cosine now the half angle for tangent can be calculated using these other two formulas it's also equal to one minus cosine divided by sine and is equal to sine divided by one plus cosine use the half-angle formula to evaluate cosine of 15 degrees now here's the formula cosine theta divided by 2 is equal to plus or minus the square root of one plus cosine theta over two so theta divided by two is equal to fifteen so if we set 15 equal to theta divided by two and if we multiply both sides by 2 we can see that 30 is equal to theta so we need to plug in 30 into this angle here so cosine 15 is plus or minus the square root of 1 plus cosine 30 divided by 2. now we know the value of cosine 30. cosine 30 is equal to the square root of 3 over 2. so inside the square root i'm going to multiply the top and the bottom by 2 to simplify the expression that we have so first we have 2 times 1 which is simply 2. and then two times the square root of three over two the twos will cancel and you're just going to get the square root of three and on the bottom we have a four now the square root of four is two so this simplifies to the square root of 2 plus the square root of 3 divided by 2. now 15 degrees is in what quadrant quadrant 1 2 3 or 4. 15 is an acute angle in quadrant one cosine is positive in quadrant one so we have two sines we're going to choose the positive sign based on the quadrant so the final answer is positive square root two plus square root three divided by 2. now if you wish to confirm your answer with the calculator you can do so if you get the decimal value of this thing if you plug it in your calculator you should get .9659 now some calculators if you type in cosine of 15 degrees you might get an answer that looks like this square root 6 plus square root 2 divided by 4. it turns out that these two things they have the same decimal value so just want to point that out now let's work on another example find the exact value of sine 22.5 degrees so feel free to pause the video now let's write the formula first sine theta divided by 2 is plus or minus the square root of 1 minus cosine theta over two so in this case we can see that 22.5 is equal to theta over two now if that's the case let's multiply both sides by two so the twos here will cancel and 22.5 times two is forty-five therefore we gotta plug in an angle of forty-five degrees into this expression so sine of 22.5 degrees is equal oh by the way is sine gonna be positive or negative 22.5 is in quadrant one so sine is positive in quadrant so we're going to use just the positive sign of that equation so sine 22.5 is positive square root 1 minus cosine 45 divided by 2. and cosine 45 is square root 2 divided by 2. so now let's multiply everything the top and the bottom by 2 inside the radical so this is going to be 2 times 1 which is simply 2 and then these twos will cancel leaving behind negative square root 2 and on the bottom 2 times 2 is 4. now the last thing that we can do is take the square root of 4. the square root of four is two so our final answer is the square root of two minus root two over two and so that's it now let's get the decimal value of that expression this is equal to 0.3827 if you type in sine 22.5 it gives you this value as well let's try this one tangent of 75 degrees go ahead and evaluate now i'm going to use this particular formula tangent theta divided by 2 is 1 minus cosine theta over sine theta now if theta over 2 is 75 then the angle theta is 75 times 2 or 150 so tan 75 is 1 minus cosine of 150 degrees divided by sine of 150 degrees now cosine of 150 that has a reference angle of 30 cosine 30 is the square root of three over two but cosine 150 150 is in quadrant two so cosine is negative so we're gonna have negative square root of 3 over 2. sine 150 is positive a half so this is 1 plus square root 3 over 2 divided by 1 half now let's multiply the top and the bottom by 2. so let's distribute the two on top two times one that's going to be two and then two times the square root of three over two the twos will cancel leaving behind the square root of three and two times a half is one so the final answer is two plus the square root of three and so that's the value of tangent 75 degrees that is the exact value you