find all the values of x that satisfy the equation: 2sin^2(x) + 1 = 3sin(x).
2sin^2(x) - 3sin(x) + 1 = 0
(2sinx - 1)(sinx - 1) = 0
sinx = 1/2 or sinx = 1
In [0,2π)
x = π/6 or 5π/6 or π/2
To find the values of x that satisfy the equation 2sin^2(x) + 1 = 3sin(x), we can rewrite it in a quadratic form.
Let's start by rearranging the equation:
2sin^2(x) - 3sin(x) + 1 = 0
This is now a quadratic equation in terms of sin(x). We can solve it using factoring, completing the square, or using the quadratic formula. We will use factoring in this case.
To factor the equation, we need to find two numbers that multiply to give 2 and add to give -3. The factors that satisfy these conditions are -2 and -1.
So, we can rewrite the equation factoring the quadratic:
(2sin(x) - 1)(sin(x) - 1) = 0
Now, we can set each factor equal to zero and solve for x:
2sin(x) - 1 = 0 or sin(x) - 1 = 0
For the first equation, we isolate sin(x):
2sin(x) = 1
sin(x) = 1/2
The possible solutions for sin(x) = 1/2 are x = π/6 and x = 5π/6 (using the unit circle or reference angles).
For the second equation, we isolate sin(x):
sin(x) = 1
The only possible solution for sin(x) = 1 is x = π/2.
Therefore, the values of x that satisfy the equation 2sin^2(x) + 1 = 3sin(x) are x = π/6, 5π/6, and π/2.
To find all the values of x that satisfy the equation 2sin^2(x) + 1 = 3sin(x), we can rearrange the equation and solve for sin(x).
1. Start by subtracting 3sin(x) from both sides of the equation to get:
2sin^2(x) - 3sin(x) + 1 = 0
2. Notice that this quadratic equation is in terms of sin(x). Let's simplify it further by factoring:
(sin(x) - 1)(2sin(x) - 1) = 0
3. Now, set each factor equal to zero and solve for sin(x):
sin(x) - 1 = 0 or 2sin(x) - 1 = 0
Solving the first equation gives:
sin(x) = 1
Solving the second equation gives:
2sin(x) = 1
sin(x) = 1/2
4. To find the values of x, we need to determine the angles at which sin(x) equals 1 and sin(x) equals 1/2. These angles can be found using the unit circle or a calculator.
For sin(x) = 1, the angle x is pi/2 + 2pi*n, where n is an integer.
For sin(x) = 1/2, the angle x is pi/6 + 2pi*n or 5pi/6 + 2pi*n, where n is an integer.
Therefore, the values of x that satisfy the equation are:
x = pi/2 + 2pi*n or x = pi/6 + 2pi*n or x = 5pi/6 + 2pi*n
Here, n represents any integer value that allows us to generate all possible solutions for x.