in this video we're going to talk about the sum and difference formulas and how to use them in a particular problem so let's start with the sine function sine a plus b is equal to sine a cosine b plus cosine a sine b and there's another one sine a minus b is equal to sine a cosine b minus cosine a sine b so make sure you know these two formulas but let's use the first one so let's say if we want to find the value of sine 75 degrees how can we do that how can we use this formula to evaluate sine 75 so you need to ask yourself what two angles that are very common that adds to 75. two common angles on a unit circle would be 30 and 45 30 plus 45 adds to 75 so we're going to say a is 30 and b is 45 so this is going to be equal to sine of 30 cosine 45 plus cosine 30 times sine 45 so what's sine of 30 sine 30 is one half based on the unit circle cosine 45 is the square root of 2 divided by 2. cosine 30 is square root 3 over 2 and sine 45 is root 2 over two so one times root two is simply root two two times two is four radical three times radical two is square root six and these two will give us four so now we can combine this into a single fraction so the answer is radical two plus radical six divided by four this is it now what about this one let's say if we want to evaluate sine of 15 degrees what formula should we use should it be a plus b or a minus b sine 15 is the same as sine 45 minus 30 so we want to use the sine a minus b formula so sine a minus b is equal to sine a cosine b minus cosine a sine b so sine 45 minus 30 so a is 45 b is 30 this is equal to sine of 45 times cosine of 30 minus cosine 45 sine 30 sine 45 is root two divided by two cosine thirty is the square root of three over two cosine forty five root two over two and sine thirty is one half square root two times the square root of three is the square root of six two times two is four minus root two over four so the final answer is radical six minus radical two divided by four so that's it by the way if you're not going to have access to the unit circle it might be useful for you to know two special triangles the 30-60-90 triangle and the 45 90 triangle across the 30 is one across the 63 across the 92 across 45 is one across 90 is root 2. so let's say if you want to find sine 30 which you'll need to be able to do for these types of problems according to sohcahtoa's sine it's going to be equal to the opposite side opposite to 30 is one divided by hypotenuse which is opposite to the 90 degree angle and that's two so therefore sine 30 is one over two sine 60 is going to be opposite to 60 is root three and the hypotenuse is two so it's uh root three divided by two now let's say if we want to evaluate cosine thirty according to toa c a h cosine is equal to the adjacent side divided by the hypotenuse so that's root 3 over 2. now if we wish to evaluate let's say tangent 30 that's the toa part of sohcahtoa so tangent is equal to the opposite side opposite of 30 is 1 divided by the adjacent side now for tangent you're going to have to rationalize so once you rationalize the denominator it's going to be root 3 over 3 that's tangent 30. now let's use the 45 degree the 45 45 90 triangle let's say if we want to evaluate sine 45 that's going to be opposite divided by the hypotenuse so 1 over root 2 and if you multiply the top and bottom by root 2 it's going to be root 2 over 2. so that's what you can get these values from if you know these two triangles and if you know how to apply them so make sure you're familiar with this expression sohcahtoa so what it means is that sine of the angle is equal to the opposite side relative to the angle divided by the hypotenuse the cup part means that cosine theta is equal to the adjacent side divided by the hypotenuse and tangent theta is the ratio between the opposite side and the adjacent side now what about cosine 7 pi divided by 12 how can we evaluate this function what's the first thing that you would do so first let's convert the angle from radians to degrees to do that we need to multiply by 180 degrees over pi so you want to do it in such a way that the units pi will cancel or the terms fine so what's 7 times 180 divided by 12 if you don't have a calculator here's what you want to do let's break up 180 into 18 times 10 and 12 is 6 times 2. 18 is 6 times 3 10 is 5 times 2 and we could cancel a six and a two seven times three is twenty one and twenty one times five twenty times five is a hundred one times five is five so this is about 105 degrees so cosine seven pi over 12 is the same as cosine 105 degrees and 105 is the sum of two common angles that is 60 and 45 so therefore we need to use the formula cosine a plus b and this is equal to cosine a cosine b minus the sine is going to change here it's plus but it's going to switch to uh minus so it's cosine a cosine b minus sine a sine b so that's going to be a is 60 b is 45 so it's cosine 60 cosine 45 minus sine 60 sine 45 now using the triangles that we mentioned before we could tell that cosine 60 is one half cosine 45 root two over 2 sine 60 is root 3 over 2 sine 45 root 2 over 2. so this is root 2 over 4 minus root 6 over 4 which is root 2 minus root 6 divided by 4. so this is the answer let's try one more example but using a tangent function go ahead and evaluate tangent pi over 12. so like before we need to convert the angle in radians to degrees so let's multiply by 180 divided by pi so the pi values will cancel and we know that 180 is 18 times 10. 12 is 6 times 2. 18 is 6 times 3 10 is 5 times 2 so we can cancel a 6 and we can cancel a 2. leaving 3 times 5 which is 15 degrees so tangent pi over 12 is the same as tangent 15 degrees so we can tell that we need to use the tangent a minus b formula because 45 minus 30 is 15. i mean that was supposed to be a 30. so let's write the equation tangent a minus b is equal to tangent a minus so this sign stays the same minus tangent b divided by one plus this sign is opposite to whatever sign you see here so it's going to be one plus tangent a tangent b so a is 45 b is 30. so this is going to be tangent 45 minus tangent 30. divided by one plus tan forty five tan thirty so we mentioned earlier that tangent thirty is root three over three but what about tan 45 so let's draw the 45-45 90 triangle now let's focus on this 45 tangent is opposite over adjacent opposite to that angle is one the adjacent side is one so one over one is simply one so tan 45 is one so this is equal to one minus root three divided by three over one plus one times root three over three so we can get rid of this one because one times root three over three is just root three over three now what can we do to solve or simplify this expression what would you do at this point how would you simplify this complex fraction what you need to do is multiply the top and the bottom by the common denominator of those two numbers which is just three so let's distribute the three three times one is three and then minus three times root three over three the threes will cancel leaving with just root 3 and then the same thing is going to happen here 3 times 1 is 3 and then plus root 3. now can we simplify this expression further you could leave the answer like that but let's see what happens if we multiply by the conjugate of the denominator so since the denominator is 3 plus root 3 the conjugate is going to be 3 minus root 3. so on top we need to form three times three is uh nine and then we have three times negative root 3 which is negative 3 over 3 and then negative 3 3 times 3 which is another negative 3 root 3 and then finally negative root 3 times negative root 3 is positive three root three times root three is the square root of nine which is three on the bottom because they're conjugates the two middle terms will cancel three times three is nine three times negative root three is negative three root three positive root three times three is positive three root three and positive root three times negative root three is negative three so these two cancel on the bottom we just have nine minus three which is six on top we have nine plus three which is twelve and we can add these two that's going to be minus six root so at this point we could separate this into two fractions so we can divide the 12 by 6 and the 6 root 3 by 6. twelve divided by six is equal to uh two six divided by six is one we could ignore the one so it's just gonna be root three so this is the final answer and it's much more simplified than the last answer that we have so it's a 2 minus root 3. so it's clear to see that the sum and difference identity with the tangent ratio is a lot more difficult to work with than the sine and cosine functions therefore it's wise to try another example so you can get used to working with tangent so try this one go ahead and evaluate tangent 23 pi over 12. so feel free to pause the video and work on this example so let's begin by converting this into degrees so let's multiply by 180 over pi so these two will cancel and it's going to be 23 times 18 times 10 and 12 is 6 times 2. and as we've been doing 18 is six times three ten is five times two and let's cancel the six and the two so now we need to multiply 23 by three and by five so three times five is fifteen so we have twenty three times fifteen there's two ways in which we can do this we can multiply by hand five times three is fifteen let's carry over the one two times five is ten plus one that's eleven let's add a zero one times three is three one times two is two so this is going to be 345 so we're looking for tangent of 345 so what two angles can we use that adds up to 345 two angles that we can use are 120 and 2 25 120 plus 225 is 345. so this time we need to use the tangent a plus b formula which is going to be tangent a plus this side stays the same the first one on top so tangent a plus tangent b divided by one minus the one on the bottom is going to be opposite to whatever we see here so it's 1 minus tan a times tan b so a is going to be 120 b is 225 so what's tangent 120 120 is over here in quadrant two in the unit circle so that's 120 and 180 is on the negative x axis so the difference between 180 and 120 is 60. so the reference angle is 60. so using the 30-60-90 triangle we could find out tan 60 which will help us to calculate tan 120. so across the 30 is 1 across the 60's root 3 across the 90 is 2. so tangent 60 opposite to 60 is root 3 adjacent is 1. so tangent is opposite over json that's root 3 over 1. so tan 60 is root 3. now tangent is positive in quadrant one but it's negative in quadrant two so tangent 120 is therefore negative root three now what about tangent 225 225 is in quadrant four and therefore the reference angle which is the angle inside the triangle between the hypotenuse and the x-axis that angle is 45 and we know that tangent 45 is one now tangent is positive in quadrant three so tan 225 is also positive 1. so now we can use the formula tan a or tangent 120 is negative root 3 tangent b or tangent 225 is positive 1 divided by one minus tan a which is a negative root three times tan b which is one so this is the same as one minus root three and on the bottom we have two negatives so that's one plus root three now we don't have any complex fractions to deal with all we need to do at this point is multiply the top and bottom by the denominator or by the conjugate of the denominator which is 1 minus 3 3. so if you see a plus sign change it to a minus sign if you want to use the conjugate whatever you do to the bottom you must also do to the top so let's foil uh the two factors on top one times one is one one times negative root three that's minus three three and then these two will also produce minus three three and negative root three times negative root three is positive three on the bottom the two middle terms will cancel because these two are conjugates of each other so we only need to multiply the first two terms one and one and the last two terms root three times negative root three which is negative three so now we can add one plus three so that's four negative one root three minus one root three is negative two three three and on the bottom 1 minus 3 is negative 2. so let's write this as two separate fractions so 4 divided by negative 2 negative 2 3 divided by negative 2. so positive 4 divided by negative 2 is negative 2 and negative 2 root 3 divided by negative 2 is positive root 3. so the final answer is root 3 minus 2.