Sum and Difference Formulas

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
sin(x-y) = sin(x)cos(y) - cos(x)sin(y)


cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)


tan(x+y) = [tan(x) + tan(y)] / [1-tan(x)tan(y)]
tan(x-y) = [tan(x) - tan(y)] / [1+tan(x)tan(y)]


These formulas are used to find the exact value of something that isn't on the trig chart, like cos(15°).You will use only angles on the trig chart to plug into the formulas. (30°,45°,60°,90°, and any angle that has a reference angle on the trig chart)


Ex. 1cos(15°)

45-30 = 15

cos(45-30) = cos(45)cos(30) + sin(45)sin(30)

cos(15°) = (√(2)/2)*(√(3)/2) + (√(2)/2)*(1/2)

cos(15°) = (√(6)/4) + (√(2)/4)

cos(15°) = [√(6)+√(2)] /4


Ex. 2tan(75°)

45 + 30 = 75

tan(45+30) = [tan(45) + tan(30)] / [1 - tan(45)tan(30)]

tan(75) = [1 + (√(3)/3)] / [1 - (1)*(√(3)/3)]

tan(75) = [(3+√(3))/3)] / [(3 - √(3))/3)]

tan(75) = [9+√(3)] / [9-√(3)]

tan(75) = [81 + 18√(3) + 3] / [81-3]

tan(75) = [84 + 18√(3)] / [78]

tan(75) = [14 + 3√(3)] / [13]


Ex. 3sin(105°)

45+60 = 105

sin(45+60) = sin(45)cos(60) + cos(45)sin(60)

sin(105) = (√(2)/2)*(1/2) + (√(2)/2)*(√(3)/2)

sin(105) = (√(2)/4) + (√(6)/4)

sin(105) = [√(2)+√(6)] /4