20sin^2 (X)-5=0

Question

Find all exact solutions on the interval 0≤x<2π. If there is more than one answer, enter them as a comma separated list. 20sin2(x)−5=0

Answer 1

first solve the equation for sin(x).

20 sin^2(x) - 5 = 0 ==> sin(x) = +/- 1/2.

there are 4 solutions in [0, 2pi) where sin x = +/- 1/2: pi / 6, 5pi / 6, 7pi / 6, and 11 pi / 6. Normally one would take each solution and add all integer multiples of 2pi to them to generate the entire solution set. In this case we can abbreviate our answer a little bit, since 7pi / 6 is just pi / 6 + pi and, similarly, 11pi / 6 is 5pi / 6 + pi, so that the full solution set is given by x = pi / 6 + k*pi, k= 0, +/- 1, +/- 2, ... or x = 5pi / 6 + k*pi, k= 0, +/- 1, +/- 2, ....