how many solutions are there to
csc^2x=25 (0≤x<2π)
since csc x can be ±, there will be one solution in each quadrant.
To find the number of solutions to the equation csc^2x = 25 in the given interval (0 ≤ x < 2π), we can start by solving for x.
csc^2x = 25
Taking the reciprocal of both sides, we get:
sin^2x = 1/25
Taking the square root of both sides, we have:
sinx = ± 1/5
We know that sinx = y has solutions in the unit circle between -1 and 1. So, we need to find the values of x where sinx = ± 1/5 within the given interval (0 ≤ x < 2π).
First, let's find the values of x where sinx = 1/5:
x = arcsin(1/5) ≈ 0.201 rad
x = π - arcsin(1/5) ≈ 2.941 rad
Next, let's find the values of x where sinx = -1/5:
x = π - arcsin(-1/5) ≈ 0.942 rad
x = 2π + arcsin(-1/5) ≈ 5.342 rad
We can see that all the values of x are within the given interval (0 ≤ x < 2π).
Therefore, there are 4 solutions to the equation csc^2x = 25 in the interval (0 ≤ x < 2π).
To find the solutions to the equation csc^2x = 25 in the given interval, we can start by manipulating the equation to a more familiar trigonometric form.
Recall that the reciprocal of sine is the cosecant function: csc(x) = 1/sin(x). Squaring both sides of the equation, we get:
(1/sin^2x) = 25
Now, we can convert the equation to a form involving sine:
1 = 25sin^2x
dividing both sides by 25, we get:
sin^2x = 1/25
Taking the square root of both sides, we have:
sin(x) = ± √(1/25) = ± 1/5
Now, we need to find the values of x between 0 and 2π whose sine is either positive or negative 1/5. Remember that sine is positive in the first and second quadrants and negative in the third and fourth quadrants.
In the first quadrant, the principal angle whose sine is 1/5 is sin^(-1)(1/5). Similarly, in the second quadrant, the angle whose sine is 1/5 is π - sin^(-1)(1/5).
To find the angles in the third and fourth quadrants, we need to remember that sine has a period of 2π. So, in the third quadrant, the angle whose sine is -1/5 is π + sin^(-1)(1/5), and in the fourth quadrant, it is 2π - sin^(-1)(1/5).
To summarize, the angles that satisfy the equation csc^2x = 25 in the interval (0, 2π) are:
x₁ = sin^(-1)(1/5) (approximately 0.2014 radians or 11.536 degrees)
x₂ = π - sin^(-1)(1/5) (approximately 2.9404 radians or 168.464 degrees)
x₃ = π + sin^(-1)(1/5) (approximately 3.3418 radians or 191.536 degrees)
x₄ = 2π - sin^(-1)(1/5) (approximately 5.7418 radians or 328.464 degrees)
Therefore, there are four solutions to the equation csc^2x = 25 in the interval (0, 2π).