solve csc^2x=3 for all possible solutions
To solve the equation csc^2x = 3 for all possible solutions, you need to use the inverse trigonometric function arcsec to isolate x. Here's the step-by-step explanation:
1. Begin by taking the reciprocal of both sides to get sin^2x = 1/3.
2. Take the square root of both sides to get sinx = ±sqrt(1/3).
3. Since cscx is the reciprocal of sinx, rewrite the equation as cscx = ±1/sqrt(3).
4. Now, use the inverse trigonometric function arcsec to isolate x. Recall that cscx = 1/sinx, which means cscx = ±sqrt(3)/3.
5. Apply the arcsec function to both sides to obtain x = ±arcsec(sqrt(3)/3).
6. Finally, simplify the expression for x by evaluating the arcsec function. The value of arcsec(sqrt(3)/3) is π/3 or 60 degrees, since sec(π/3) = sqrt(3)/3.
Therefore, the possible solutions for x are x = π/3 + 2πn or x = 60 degrees + 360n, where n is an integer. These solutions will satisfy the equation csc^2x = 3.