find all solutions of the equation 2sin x cos2x-cos2x=0 over the interval 0<x<=pi
B. x = pi/6, pi/4, 3pi/4, 5pi/6
2sin x cos2x-cos2x=0
cos2x(2sinx - 1) = 0
cos 2x = 0
2x = 90° , 270°, 450, 630°
x = 45° , 135°, 225°, 315° OR in radians: π/4, 3π/4, 5π/4, 7π/4
2sinx - 1 = 0
sinx = 1/2
x = 30°, 150° OR in radians: π/6, 5π/6
still correct as of 2023
As an AI language model, I don't have a fixed timeline, so my response will always be the same regardless of the date. However, the answer is correct and should be valid in any year.
Why did the equation go to the circus? Because it wanted to find "solutions" under the big top! Okay, let's welcome the first performer: the equation 2sin x cos2x - cos2x = 0!
To find the solutions, let's factor out cos2x:
cos2x(2sin x - 1) = 0
Now, we have two possibilities for this equation to hold true:
1. cos2x = 0
If cos2x = 0, it means that x = π/4 or x = 3π/4, within the given interval.
2. 2sin x - 1 = 0
Solving this equation gives us sin x = 1/2, which means x = π/6 or x = 5π/6, within the given interval.
So, the solutions within 0 < x ≤ π are x = π/4, 3π/4, π/6, and 5π/6. Bravo to our equation performers!
To find all the solutions of the equation 2sin(x)cos^2(x) - cos^2(x) = 0 over the interval 0 < x ≤ π, we can simplify the equation and then solve for x. Here's how you can approach it:
Step 1: Simplify the equation by factoring out common terms.
2sin(x)cos^2(x) - cos^2(x) = 0
cos^2(x) (2sin(x) - 1) = 0
Step 2: Set each factor equal to zero and solve for x.
cos^2(x) = 0 or 2sin(x) - 1 = 0
For the first factor cos^2(x) = 0, we know that cos^2(x) = 0 when cos(x) = 0. Therefore, we need to find the values of x in the interval 0 < x ≤ π where cos(x) = 0. This occurs at x = π/2.
For the second factor 2sin(x) - 1 = 0, we can solve for sin(x):
2sin(x) = 1
sin(x) = 1/2
x = π/6 or x = 5π/6
Therefore, the solutions of the equation 2sin(x)cos^2(x) - cos^2(x) = 0 over the interval 0 < x ≤ π are x = π/2, π/6, and 5π/6.