How do I solve for the solutions? Find all solutions of the equation.

#csc^4 2theta-4=0#

1 Answer
Apr 26, 2018

pi/4 + 2kpi
(3pi)/4 + 2kpi
(5pi)/4 + 2kpi
(7pi)/4 + 2kpi

Explanation:

#csc^4 = 4#
#1/(sin^4 t) = 4# --> #sin^4 t = 1/4#
#sin^2 t = 1/2#
#sin t = +- 1/(sqrt2) = +- sqrt2/2#
Trig table and unit circle give 4 solutions:
a. #sin t = sqrt2/2# -->
#t = pi/4 + 2kpi#,and
#t = (3pi)/4 + 2kpi#
b. #sin t = - sqrt2/2# -->
#t = - pi/4#, or #t = (7pi)/4 + 2kpi # --> co-terminal
#t = pi - (-pi/4) = pi + pi/4 = (5pi)/4 + 2kpi#