I have to use Trigonometric Identities to find the exact value of expressions.?

Question

I apparently missed a BIG part of the Trig basics. I have to find values for example of Sin(375degrees) using things like 3/4 + 1/4 or sqrt3/4 - sqrt3/4, etc. I have no idea how to do this. I have a lot of these and an answer bank. I think if I can get on the right track I can do it!
Thanks.

Answer 1

Remember:
sin(A + B) = sinAcosB + cosAsin B
sin(375) = sin(330 + 45) = sin330cos45 + sin45cos330

sin 330 = -0.5
cos 45 = sin 45 = sqrt(2)/2
cos 330 = sqrt(3)/2

sin330cos45 + sin45cos330 =
-0.5(sqrt(2)/2) + [sqrt(3)/2]sqrt(2)/2 = -(sqrt(2)/4) + sqrt(6)/4 =
[sqrt(6) - sqrt(2)]/4

Answer 2

First reduce the angle to the range 0 to 2pi. 375 - 360 = 15 degrees. Then use known trig values and trig identites to get your answer. The trick being to use the formulas to convert the angle in question into angles you know the trig function values of. In the case of 15 degrees, it is half of 30 degrees, so you can write, using half angle formula:

sin^2(u) = (1 - cos(2u))/2 = (1 - cos(30))/2 = (1 - SQRT(3)/2)/2
sin(15) = SQRT((1 - SQRT(3)/2)/2)

some useful ones:
sin(30) =1/2
cos(30) = SQRT(3)/2
cos(60) = 1/2
tan(45) = SQRT(2)/2
etc

There are many tables of trig identities on the internet, I have include one here.