Find the exact value of the inverse trig function.?

Question

Find the exact value of the inverse trig functions.

(1) cos^-1(sin(7pi/2))

(2) tan^1(tan(2pi/3))

Answer 1

1)
Take it step by step, starting inside:
sin 7pi/2 = -1
so...
cos^-1 (-1) = pi

2)
In this case, the tan and tan^-1 just cancel each other out, so you're left with 2pi/3. However, if your interval is on 0 to 2pi, then you have a second solution... 5pi/3 (tan is negative in the 2nd and 4th quadrant.)

Answer 2

Inverse trig functions are multi-valued:
(1) cos^-1(sin(7π/2)) = (2n + 1)π

(2) tan^-1(tan(2π/3)) = 2nπ/3

n being any integer in both cases.

edit: my bad!
(2) s/b tan^-1(tan(2π/3)) = 2π/3 + nπ = (3n + 2)π/3