Expert: Steve Holleran - 5/3/2007

Question
1)tan 105 degrees
2)sin 105 degrees
Find the exact value?

Answer
Hi Chris,

The trick here is to use the Addition Formulas and to break down the 105 into angles that you can read the trig values of from diagrams.

For (1), think of tan 105 as tan(60 + 45)

Then, according to the formula for tan(A + B),

tan(60 + 45) = [tan 60 + tan 45] / [1-tan 60 * tan 45]

You can get the values by diagramming a 30-60-90 triangle and a 45-45-90 triangle.  

Remember on a 30-60-90, the side opposite the 30 is 1, the side opposite 60 is sqrt(3) and the hypotenuse is 2 (these are ratios, not actual lengths).

Similarly, in the 45-45-90, the legs are 1 and the hypotenuse is sqrt(2).

Now just use SOH-CAH-TOA and get the values:

tan(60 + 45) = [sqrt(3) + 1] / [ 1 - sqrt(3) * 1]

            = [1 + sqrt(3)] / [1 - sqrt(3)]

Do the same for (2): just use the formula
sin(A + B) = sin A cos B + cos A sin B

sin(105) = sin(60 + 45) = sin 60 cos 45 + cos 60 sin 45

        =  sqrt(3)/2 * 1/sqrt(2) + 1/2 * 1/sqrt(2)

        =  [sqrt(3) + 1] / 2 * sqrt(2)

I hope this is what you wanted.

Steve Holleran