Question

Find the inverse of:
f(x) = -5 +11arccos(7^x - 9)

Find exactly all the real solutions for each of the following equations.
a. Sin(x) = - square root 3/2
b. 1 + cos(3x-pi/4) = 1/2

Answer 1

x = -5 + 11arccos(7^y-9)
(x+5)/11 = arccos(7^y-9)
7^y-9 = cos (x+5)/11
7^y = 9 + cos (x+5)/11
y = log₇ (9 + cos (x+5)/11)

you know that sin π/3 = √3/2
so, 2kπ + π/3 or (2k+1)π - π/3
or (2k+1)(π/2)±(π/6)

1+cos(3x-π/4) = 1/2
cos(3x-π/4) = -1/2
3x-π/4 = 2π/3 or 4π/3
3x = 11π/12 or 19π/12 (+2π,4π,6π)
x = 11π/36 or 19π/36 (+2π/3,4π/3,2π)