cosxtanx-sin^2x=0
solve for x over the domain
0 is less than or equal to x which is less than or equal to 2pi
sin(x) - sin^2(x) = 0
sin(x) [1 - sin(x)] = 0
sin(x) = 0
sin(x) = 1
x= pi, 0, 2pi, and pi/2 ?
Well, well, well, looks like we have a mathematical equation in need of some fun! Let's dive in and solve it with a dash of humor!
cos(x)tan(x) - sin^2(x) = 0
First, we need to rewrite it a bit:
cos(x)tan(x) = sin^2(x)
Hold on, before we proceed, let's make a quick observation. Recall that tan(x) is equal to sin(x)/cos(x). So, we can actually rewrite our equation like this:
cos(x) * (sin(x)/cos(x)) = sin^2(x)
Now, let's get down to business and simplify things:
sin(x) = sin^2(x)
Oh, snap! This equation has a sneaky solution lurking around. Notice that when sin(x) is equal to zero, sin^2(x) is also equal to zero. That means one of our solutions is x = 0.
But wait, there's more! The equation sin(x) = sin^2(x) also holds true for sin(x) = 1, because 1 squared is still 1. So, our other solution is x = pi/2.
With our trusty domain in mind (0 ≤ x ≤ 2π), we can see that our solutions fall within the given range.
So, in conclusion, the solutions to the equation cos(x)tan(x) - sin^2(x) = 0 in the domain 0 ≤ x ≤ 2π are x = 0 and x = pi/2. Ta-da!
Remember, math can be fun too! Keep smiling and solving those equations! 🤡
To solve the equation cos(x)tan(x) - sin^2(x) = 0 over the given domain, we can follow these steps:
Step 1: Simplify the equation.
Using trigonometric identities, we can simplify the equation:
cos(x)tan(x) - sin^2(x) = 0
(cos(x) / sin(x)) * sin(x) - sin^2(x) = 0
cos(x) - sin^2(x) = 0
Step 2: Rearrange the equation.
Rearrange the equation to isolate sin^2(x):
sin^2(x) = cos(x)
Step 3: Apply the identity.
Using the Pythagorean Identity for trigonometric functions:
sin^2(x) + cos^2(x) = 1
We can replace cos(x) with 1 - sin^2(x) in the equation from Step 2:
sin^2(x) = 1 - sin^2(x)
Step 4: Solve for sin(x).
Combine like terms:
2sin^2(x) = 1
sin^2(x) = 1/2
Take the square root of both sides:
sin(x) = ±√(1/2)
Step 5: Solve for x.
To solve for x, we need to find the values of x that correspond to ±√(1/2) in the given domain of 0 ≤ x ≤ 2π.
For sin(x) = √(1/2):
Using the inverse sine function, we find:
x = arcsin(√(1/2)) = π/4
For sin(x) = -√(1/2):
Using the inverse sine function and the symmetry of the sine function, we find:
x = π - arcsin(√(1/2)) = 3π/4
So, the solutions for x in the given domain are x = π/4 and x = 3π/4.