Question

Evaluate: tan(arcsin(8/17)+ arctan(4/3))
I understand that I have to make triangles out of the values given-- with 15 being the other side of the 17 and 8 triangle, and 5 being the other side of the 4 and 3 triangle. I'm not sure where to go on from there, though.

Answer 1

Yeah, you have to draw two different triangles for them. It's easier to solve this if you draw them.
Anyway, we let arcsin(8/17) be equal to some angle, A. And we let arctan(4/3) be equal to some angle, B. In your drawing of two triangles, label these angles as A and B.
We can rewrite the expression tan(arcsin(8/17)+ arctan(4/3)) as:
tan(A + B)
Using the formula for sum of tangents, we'll have its expanded for:
( tan(A) + tan(B) ) / ( 1 - tan(A)tan(B) )
Since you have your drawing of triangles, you now put values for each. From the drawing, we know that
tan A = 8/15, and
tan B = 4/3
substituting,
= ( 8/15 + 4/3 ) / ( 1 - 8/15 * 4/3 )
= 84/13

hope this helps~ `u`

Answer 2

Ah, that makes sense! I always forget the trig addition formulas, thank you!