Find all exact solutions on the interval 0 ≤ x < 2π csc^2(x) − 6 = −2

Answer 1

csc^2(x) - 6 = 2

csc^2(x) = 4

sqrt [csc^2(x) ] = sqrt 4

csc x = ±2

sin x = ±(1/2)

x = pi/6, 5pi/6, 7pi/6, 11pi/6

Answer 2

First, let's rearrange the equation to isolate the trig:

1. csc²x - 6 = -2 becomes csc²x = 4

Next, take the square root of both sides:

1. csc(x) = 2 or csc(x) = -2 (since (2)² and (-2)² both equal 4)

csc(x) by definition is 1/sin(x)

Therefore sin(x) = 1/2 or sin(x) = -1/2

By consulting the unit circle, we know that:

1. sin(x) = 1/2 at π/6 and 5π/6
2. sin(x) = -1/2 at 7π/6 and 11π/6

Therefore, our solutions are π/6, 5π/6, 7π/6, and 11π/6