Solve for x. sec(x)sin(x)=2sin(x)
0<=x<=2pi
sin(x)sec(x) - 2sin(x) = 0
sin(x)(sec(x)-2) = 0
sin(x) = 0
sec(x) = 2
Those are pretty standard angles, so listing the solutions should be easy.
To solve for x in the equation sec(x)sin(x) = 2sin(x), we need to isolate sin(x) and then solve for x. Here's the step-by-step process:
Step 1: Divide both sides of the equation by sin(x):
sec(x)sin(x) / sin(x) = 2sin(x) / sin(x)
Since sin(x) / sin(x) is equal to 1, we get:
sec(x) = 2
Step 2: Take the reciprocal of both sides to get cosine:
cos(x) = 1 / sec(x)
Since sec(x) is the reciprocal of cos(x), we have:
cos(x) = 1/2
Step 3: Solve for x using the inverse cosine function:
x = cos^(-1)(1/2)
Step 4: Use the values of x from 0 to 2π (the given range) to find possible solutions.
In this case, cos^(-1)(1/2) is equal to π/3. The inverse cosine function returns the principal value between 0 and π, but we also need to consider the values between π and 2π.
So, the possible values for x are π/3 and 5π/3.
Therefore, the solution for x in the given range 0 <= x <= 2π is:
x = π/3 and x = 5π/3.