Determine exact solution. x E (0, 2pi)

2csc^2x-8=0

2csc^2x-8=0

csc^2 x = 4/2 = 4
then
sin^2 x = 1/4
sinx = ±1/2 ---> so x can be in any of the 4 quads
we know sinπ/6 = 1/2

so
x = π/6
x = π - π/6 = 5π/6
x = π + π/6 = 7π/6
x = 2π - π/6 = 11π/6

To find the exact solution for the equation 2csc^2(x) - 8 = 0 within the interval (0, 2pi), we can follow these steps:

Step 1: Solve the given equation for csc^2(x):

2csc^2(x) - 8 = 0

Add 8 to both sides of the equation:

2csc^2(x) = 8

Divide both sides by 2:

csc^2(x) = 4

Step 2: Take the square root of both sides:

√(csc^2(x)) = √(4)

Simplifying the square root:

|csc(x)| = 2

Step 3: Solve for x using the reciprocal identity of csc(x):

csc(x) = 2

Since the absolute value of csc(x) is 2, it means that either csc(x) = 2 or csc(x) = -2.

For csc(x) = 2:
To find the reference angle for an inverse csc function equal to 2, use the reciprocal of 2, which is 1/2. The reference angle, θ, will be sin^(-1)(1/2) = π/6 or 30 degrees.

Since csc(x) is positive, the solution within the given interval is:

x = π/6

For csc(x) = -2:
To find the reference angle for an inverse csc function equal to -2, use the reciprocal of -2, which is -1/2. The reference angle, θ, will be sin^(-1)(-1/2) = 7π/6 or 210 degrees.

Since csc(x) is negative, the solutions within the given interval are:

x = 7π/6 or x = 11π/6

Therefore, the exact solutions for the equation 2csc^2(x) - 8 = 0, within the interval (0, 2π), are:

x = π/6, 7π/6, 11π/6

To find the exact solution of the given equation, we'll follow these steps:

Step 1: Start with the given equation:
2csc^2x - 8 = 0

Step 2: Rearrange the equation to isolate the trigonometric term:
2csc^2x = 8

Step 3: Take the reciprocal of both sides of the equation to remove the squared term:
csc^2x = 1/4

Step 4: Rewrite the equation using the identity csc^2x = 1/sin^2x:
1/sin^2x = 1/4

Step 5: Cross-multiply to get rid of the fractions:
1 * 4 = sin^2x * 1
4 = sin^2x

Step 6: Take the square root of both sides of the equation:
√4 = √sin^2x
2 = |sinx|

Step 7: Solve for sinx by considering both the positive and negative roots:
sinx = 2 OR sinx = -2

Step 8: However, since sinx can only have values between -1 and 1, there is no solution to sinx = 2 or sinx = -2.

Therefore, the given equation 2csc^2x - 8 = 0 has no exact solutions in the interval (0, 2pi).