csc^2 x = 5 over the interval 0 <= x < 2pi

1. 26.57 degs

2. 153.43 degs

I think the solutions in the third and fourth quadrants don't work because they don't fall in the interval.

csc^2 x = 5

or
sin^2 x = 1/5
sin x = ± 1/√5

So x is in all 4 quadrants.
the reference angle is 26.57°

so x = 26.57°
x = 180-26.57 = 153.43°
x = 180 + 26.57 = 206.57°
or
x = 360-26.57 = 333.43°

just notice that your domain is 0 ≤ x ≤ 2π
so you would want your answers in radians,
x = .4636
x= π - .4636 = 2.68
x = π + .4636 = 3.61
x = 2π-.4636 = 5.82

Thank you so much!

To solve the equation csc^2(x) = 5 over the interval 0 <= x < 2π, we can start by taking the reciprocal of both sides of the equation to get sin^2(x) = 1/5.

Then, we can take the square root of both sides to get sin(x) = ± √(1/5).

Now, we need to find the values of x in the interval 0 <= x < 2π that satisfy sin(x) = √(1/5) and sin(x) = -√(1/5).

To find these values, we can use the unit circle or a calculator to determine the angles whose sine values are equal to √(1/5) and -√(1/5).

Using a calculator, we find that sin^(-1)(√(1/5)) = 26.57 degrees (approximately) and sin^(-1)(-√(1/5)) = -26.57 degrees (approximately).

Since the interval is given as 0 <= x < 2π, we convert the angles to radians by multiplying by π/180, giving us approximately 0.464 radians for 26.57 degrees and -0.464 radians for -26.57 degrees.

Therefore, the solutions in the given interval are approximately 0.464 radians (26.57 degrees) and -0.464 radians (-26.57 degrees).

So, the correct answer would be:

1. 26.57 degrees (approximately) or 0.464 radians (approximately)