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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 to 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 the sine of 30 cosine 45 plus cosine 30 times sine 45 so what's sine of 30 sine 30 is 1/2 based on a 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 2 so 1 times root 2 is simply root 2/2 times two is for 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 2 plus radical 6 divided by 4 this is it now what about this one let's say if we want to evaluate sine of 15 degrees what formulas 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 former 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 the sine of 45 times cosine of 30 minus cosine 45 sine 30 sine 45 is root 2 divided by 2 cosine 30 is the square root of 3 over 2 cosine 45 root 2 over 2 and sine 30 is 1/2 square root 2 times the square root of 3 is the square root of 6 2 times 2 is 4 minus root 2 over 4 so the final answer is radical 6 minus radical 2 divided by 4 so that's it by the way if you're not going to have access to the inner circle it might be useful for you to know two special triangles the 30-60-90 triangle and the 45-45-90 triangle across the 30 is one across the 60 root 3 across the ninety two 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 sine is going to be equal to the opposite side opposite to 30 is 1 / hypotenuse which is opposite to the 90-degree angle and that's 2 so therefore sine 30 is 1 over 2 sine 60 is going to be opposite to 60 is root 3 and the hypotenuse is 2 so it's a root 3 divided by 2 now let's say if we want to evaluate cosine 30 according to sohcahtoa cos CAH 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 meet with 2 over 2 so that's why 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 Club 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 unit's PI will cancel or the terms fine so what's seven times 180 divided by 12 if you don't have a calculator here's what we 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 6 and a 2 7 times 3 is 21 and 21 times 5 20 times 5 is 100 1 times 5 is 5 so this is about 105 degrees so cosine 7 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 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 1/2 cosine 45 root 2 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 it to leave in 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 1 plus this sine is opposite to whatever sign you see here so it's going to be 1 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 1 plus tan 45 10 30 so we mentioned earlier that tangent 30 is root 3 over 3 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 angles 1 the adjacent side is 1 so 1 over 1 is simply 1 so tan 45 is 1 so this is equal to 1 minus root 3 divided by 3 over 1 plus 1 times root 3 over 3 so we can get rid of this 1 because 1 times root 3 over 3 is Jeff's root 3 over 3 now what can we do to solve or simplify in 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 string so let's distribute the 3 3 times 1 is 3 and then minus 3 times root 3 over 3 the 3s will cancel leave in 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 3 times 3 is 9 and then we have 3 times negative root 3 which is negative 3 or 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 3 root 3 times root 3 is the square root of 9 which is 3 on the bottom because they're conjugates the two middle terms will cancel 3 times 3 is 9 3 times negative root 3 is negative 3 root 3 positive root 3 times 3 is positive 3 root 3 and positively 3 times negative 3 is negative 3 so these two cancel on the bottom we just have 9 minus 3 which is 6 on top we have 9 plus 3 which is 12 and we can add these two that's going to be minus 6 3 3 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 12 divided by 6 is equal to 2 6 divided by 6 is 1 we could ignore the 1 so it's just going to be root 3 so this is the final answer and it's much more simplified than the last answer that we have so it's 2 minus 3 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 then 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 6 times 3 10 is 5 times 2 and let's cancel the 6 and the 2 so now we need to multiply 23 by 3 and by 5 so 3 times 5 is 15 so we have 23 8 times 15 there's 2 ways in which we can do this we can multiply by here 5 times 3 is 15 let's carry over the 1 2 times 5 is 10 plus 1 at 11 let's add a 0 1 times 3 is 3 1 times 2 is 2 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 225 120 plus 225 is 345 so this time we need to use these tangent a plus B formula which is going to be tangent a plus this sign 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 can find out tan 60 which will help us to calculate 10 120 so across the 30 is 1 across the 60 is root 3 across the 90 is 2 so tangent 60 opposite to 60 is root three adjacent is one so tangent is opposite over adjacent that's root three over one so tan 60 is root three 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 4 and therefore the reference angle which is the angle inside the triangle between the hypotenuse and the x-axis that angle is 45 now we know that tangent 45 is 1 now tangent is positive in Quadrant dream so tan 225 is also positive 1 so now we can use the formula tan a or tangent 120 is a negative root 3 tangent B or tangent 225 is positive 1 divided by 1 minus tan a which is a negative root 3 times tan B which is 1 so this is the same as 1 minus root 3 and on the bottom we have two negatives so that's 1 plus root 3 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 the 2 factors on top 1 times 1 is 1 1 times negative root 3 that's minus root 3 and then these two will also produce minus root 3 and negative root 3 times negative 3 3 is positive 3 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 in one and the last two terms root 3 times negative root 3 which is negative 3 so now we can add one plus three so that's 4 negative 1 root 3 minus 1 root 3 is negative 2 3 3 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 and 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