Sum to Product Identities and Product to Sum Formulas Trig Examples

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The Organic Chemistry Tutor

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in this video we're gonna talk about how to use the sums of product formulas and the product to some identities so let's begin let's say if you want to find the exact value of cosine 195 this is in degrees plus cosine 105 how can you do it so the formula that you need is this one cosine a plus cosine B is equal to two cosine a plus B divided by two times cosine a minus B divided by two by the way you can always go to Google Images and look up the sums of product formulas and the products of sum formulas this particular formula is a sum to product formula as you can see we have a sum of two cosines on the left and on the right we have the product of two cosines these are multiplied to each other so let's use the formula we need to realize that a is 195 and B is 105 so close sine a plus cosine B is equal to two cosine a plus B that's 195 plus 105 divided by two times cosine a minus B or 195 minus 105 divided by two so 195 plus 105 and that's three hundred and three hundred divided by two is 150 degrees 195 minus 105 and that's about 90 if my math is correct and ninety divided by two is 45 so now what is cosine 150 and what's cosine 45 let's start with cosine 45 we can figure that out if you know the 45-45-90 triangle across the 45 angles it's one across the 90 is root 2 so cosine 45 used in sohcahtoa cosine is the adjacent side divided by the hypotenuse so relative to this 45 the adjacent side is 1 the hypotenuse is root 2 so if you rationalize it cosine 45 is root 2/2 now what about cosine 150 what is the value of that unless you have it memorized or unless you have a unit circle in front of you here's what you can have to do to figure it out draw the angle in 150 150 is in Quadrant 2 so this is a hundred and fifty degrees now the negative x-axis is 180 the difference between 180 and 150 is the reference angle 30 now the reason why you want to find a reference angle is because you can evaluate cosine 30 which is very similar to cosine 150 using the 30-60-90 triangle across the 30 is 1 across the 60 is root 3 across the 90s 2 so cosine of 30 which is the reference angle is equal to the adjacent side root 3 divided by the hypotenuse 2 now cosine 150 is also going to be root 3 over 2 but we need to find out if it's going to be positive or negative cosine is associated with the x value and X is negative in Quadrant 2 because it's towards the left so cosine 150 is negative root 3 over 2 so this is 2 times a negative root 3 over 2 so the final answer is going to be we can cancel a 2 radical 3 times radical 2 is radical 6 so the final answer is negative root 6 over 2 let's try another example try this one what is the exact value of sine 75 degrees plus sine 15 degrees what would you do to get the answer so we need to use the formula sine a plus sine B is equal to 2 sine a plus B divided by 2 times cosine a minus B divided by 2 so in this problem a is gonna be 75 and B is 15 so sine a plus sine B is going to be 2 times sine a plus B or 75 plus 15 divided by 2 and cosine a minus B or 75 minus 15 divided by 2 so 75 plus 15 is 90 and half of 90 or 90 divided by 2 is 45 now 75 minus 15 is 60 and 60 divided by 2 is 30 sine 45 is the same as cosine 45 it's going to be root 2 over 2 now cosine 30 we got that answer and the last problem using the 30-60-90 triangle so we know it's root 3 over 2 so once again we can cancel it - root 2 times root 3 is the square root of 6 over 2 so this is the answer now instead of evaluating sine 75 plus sine 15 which is the sum what if we have to evaluate a product of two sine functions with different angles let's say sine 1 35 times sine 75 now we know the value of sine 1 35 using the unit circle but we don't know what sine 75 is so we're still going to use the product to sum formula as you can see we have a product of two signs so the equation that we need is this one sine a times sine B is equal to 1/2 times a cosine a minus B minus cosine a plus B so this type of formula is called product to sum so a is 135 and B is 75 so therefore sine a times sine B is going to be 1/2 cosine a minus B or cosine 135 minus 75 and then minus cosine a plus B or cosine 135 plus 75 so what is 135 minus 75 that's gonna be 60 and what about 135 plus 75 that's 210 so now what is cosine 60 and what's cosine 210 cosine 60 is 1/2 cosine 210 is very similar to cosine 30 because 210 produces a reference angle of 30 210 is in Quadrant 3 and is 30 degrees away from 180 the negative x-axis so we know cosine 30 is root 3 over 2 but cosine 210 because it's in Quadrant 3 and X is negative in that region cosine 210 is negative root 3 over 2 so now let's do the math so this is 1/2 x now the two negative signs will turn into a positive sign so 1/2 plus root 3 over 2 now we can combine these two fractions into a single fraction so we can write it as 1 plus root 3 divided by 2 now 1 times the numerator will stay the same it's not going to have any effect on it and 2 times 2 is 4 so the final answer is 1 plus root 3 divided by 4 so this is it now what about this example what's sine 15 times cosine 105 go ahead and use the product to sum formulas to evaluate this function to find the exact value so what equation do we need here's the equation that you need sine a cosine B is equal to 1/2 cosine actually it's 1/2 sine a plus B plus sine a minus B so a is 15 and B is 105 so sine 15 times cosine 105 that's gonna be 1/2 sine a plus B or 15 plus 105 plus sine a minus B or 15 minus 105 15 plus 105 that's 120 and 15 minus 105 that's going to be negative 90 now sine is an odd function sine negative X is equal to negative sine X cosine is an even function so we can rewrite this as sine 120 minus sine 90 now what's sign 120 sign 120 is the same as sine 60 and sine 60 is root 3/2 if you use the 30-60-90 triangle and in quadrants one and two sine is positive since Y is positive sine 90 is one so let's get common denominators one is the same as two over two so we can combine this into a single fraction and write it as radical 3 -2 divided by 2 so x 1/2 the final answer is going to be radical 3 -2 divided by 4 so this is it so now you know how to use the sums of product formulas and the products of some formulas thanks for watching

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