# Product To Sum Identities and Sum To Product Formulas – Trigonometry

product to sum identities
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now lets review the product to seven formulas heres the first one sine alpha cosine beta is equal to 1/2 times cosine alpha minus beta minus cosine alpha plus beta so thats the first one you need to know now lets write the other three the next one is cosine alpha times cosine beta and thats equal to 1/2 cosine alpha minus beta and this time were gonna have a positive sign instead of a negative sign so plus cosine alpha plus beta and so thats it for the second equation now the third one is this one sine alpha cosine beta is equal to 1/2 now this time were going to be using sine instead of cosine so its sine alpha plus beta plus sine alpha minus beta and finally the last one we need to note is cosine alpha sine beta which is equal to 1/2 sine alpha plus beta minus sine alpha minus beta so those are the four equations they need now lets use the product to some formulas to simplify trigonometric expressions so lets say we have sine 7x x sine 4x how can we simplify this expression now if you recall sine alpha times sine beta is equal to 1/2 cosine alpha minus beta minus cosine alpha plus beta so in this example notice that alpha is 7x beta is 4x so using the formula its going to equal 1/2 cosine alpha minus beta or 7x minus 4 X minus cosine alpha plus beta which is 7 X + 4 X 7x minus 4x is just reacts and 7x plus 4x is 11 X so therefore this is the answer sine 7x times sine 4 X is equal to 1/2 cosine 3x minus cosine 11 X lets try another example sine 9 x times cosine 3x feel free to pause the video and use the product to sum formula to simplify this expression so lets begin with the equation sine alpha cosine beta thats equal to 1/2 sine alpha plus beta plus sine alpha minus beta so go ahead and use that equation and simplify the expression so first we need to identify the two angles alpha is 9x beta is 3x so this is going to be equal to 1/2 sine 9x plus 3x plus sine 9x minus 3x so 9x plus 3x thats 12x 9x minus 3x is 6x so here we have the final answer sine X cosine 3x is equal to what we have here now lets go over the sum to product formulas that you need to know so the first one is this sine alpha plus sine beta so we have a sum of two trig functions this is equal to two sine alpha plus beta divided by two times cosine alpha minus beta over 2 so if you need to add two sine values with different angles you can use that formula now the next one sine alpha minus sine beta so this is equal to two sine alpha minus beta divided by two times cosine alpha plus beta divided by two number three cosine alpha plus cosine beta so thats equal to two cosine alpha plus beta this is basically the average of the two angles and then times cosine alpha minus beta divided by two now the last one cosine alpha minus cosine beta thats equal to negative 2 its a little different the last ones negative 2 sine alpha plus beta divided by 2 times sine alpha minus beta over 2 so those are the four formulas that you need to know lets work on this example sine 8x plus sine 3x lets simplify this expression so first lets write the form of sine alpha plus sine beta is equal to 2 sine alpha plus beta divided by 2 multiplied by cosine alpha minus beta divided by 2 so alpha is 8x and beta mystery X so then this is going to be 2 sine alpha plus beta 8 X plus 3x divided by 2 times cosine alpha minus beta 8 X minus 3x divided by 2 8 X plus 3x thats 11 X + 8 X and minus 3x is 5x so therefore sine 8 X plus sine 3x is equal to 2 sine 11 X divided by 2 times cosine 5x divided by 2 lets try another example cosine 11x plus cosine 3x use the sum to product formula to simplify this expression so lets write the equation that we need first cosine alpha plus cosine beta thats equal to 2 cosine alpha plus beta divided by 2 x cosine alpha minus beta over 2 so now you want to identify what alpha is equal to so we can see that alpha is 11x and lets make beta 3x so alpha plus beta 11 X plus 3x and on this side we have alpha minus beta thats 11 X and minus 3x so 11 X plus 3x thats 14 X + 11 X minus 3x is 8 X 14 X divided by 2 is 7x 8x divided by 2 is 4x so here is our final answer lets try this example simplify sine 75 plus sine of 15 degrees so what we have is a sum of two sine values so we got to use the sum to product formula so we know its sine alpha plus sine beta and thats equal to 2 sine alpha plus beta divided by 2 times cosine alpha minus beta divided by 2 so lets identify the angles alpha is 75 beta is 15 now lets plug in the angles into the formula so alpha plus beta thats 75 plus 15 cosine alpha minus beta thats 75 minus 15 now 75 plus 15 is 90 90 divided by 2 is 45 75 minus 15 is 60 60 divided by 2 is 30 now sine 45 is equal to the square root of 2 divided by 2 cosine 30 is the square root of 3 divided by 2 and so we can cancel the twos 2 times 3 is 6 so were going to have the square root of 6 divided by two and so that is the exact value of this expression and now lets work on a verifying identity problem lets show that sine X plus sine 3x divided by cosine X plus cosine 3x lets prove that this is equal to tangent on top we have the sum of two sine functions so therefore we need to use the sum to product formula which is sine alpha plus sine beta is equal to two sine alpha plus beta divided by two times cosine alpha minus beta divided by two so alpha Im gonna choose the larger angle to be alpha thats three X and Im gonna make the smaller one beta so thats going to be equal to two sine alpha plus beta thats three X plus X divided by two times cosine alpha minus beta three X minus X divided by two now on the bottom we have the sum of two cosine values cosine alpha plus cosine beta thats equal to two cosine alpha plus beta divided by two times cosine alpha minus beta divided by two hopefully your teacher will give you a formula sheet so you have to memorize these equations because that will be a difficult thing to do or at least its gonna be time-consuming now cosine X plus cosine 3x thats going to be two cosine alpha plus beta three X plus X divided by 2 and cosine alpha minus beta thats going to be three X minus X divided by two so these two expressions are exactly the same so therefore we dont need them around we could cancel it three X plus X is 4x and 4x divided by 2 is 2x we could also cancel the two so left over with sine 2x divided by cosine 2x and sine divided by cosine is tangent so therefore we verified the identity tags:
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