Find all values of x in the interval [0, 2π] that satisfy the equation.
6 sin2(x) = 3
You want sin(x) = 1/sqrt(2) or -1/sqrt(2)
So, select all multiples of π/4 between 0 and 2π
To find all values of x that satisfy the equation 6 sin^2(x) = 3 in the interval [0, 2π], we can follow these steps:
Step 1: Divide both sides of the equation by 6 to isolate sin^2(x):
sin^2(x) = 3/6
sin^2(x) = 1/2
Step 2: Take the square root of both sides of the equation to solve for sin(x):
sin(x) = ±√(1/2)
sin(x) = ±(√2 / 2)
Step 3: Identify the possible values of x in the interval [0, 2π] that correspond to the value of sin(x) calculated in the previous step.
When sin(x) = √2 / 2, there are two possible values: π/4 and 3π/4.
When sin(x) = -√2 / 2, there are two possible values: 5π/4 and 7π/4.
Therefore, the values of x in the interval [0, 2π] that satisfy the equation 6 sin^2(x) = 3 are:
x = π/4,
x = 3π/4,
x = 5π/4, and
x = 7π/4.
To find all the values of x in the interval [0, 2π] that satisfy the equation 6 sin^2(x) = 3, we can start by rearranging the equation:
sin^2(x) = 3/6
sin^2(x) = 1/2
Taking the square root of both sides gives:
sin(x) = √(1/2)
Now, we need to find the values of x that make sin(x) equal to √(1/2) within the given interval [0, 2π].
Since sin(x) is positive in the first and second quadrants, we can focus on finding the values of x in those quadrants that make sin(x) equal to √(1/2).
In the first quadrant (0 to π/2), sin(x) is positive and √(1/2) is also positive. So, there is one solution in this quadrant.
In the second quadrant (π/2 to π), sin(x) is positive, but √(1/2) is negative. However, since sin(x) is an odd function, sin(x) = √(1/2) is equivalent to sin(x) = -√(1/2), which has another solution in this quadrant.
Using the inverse sine function (arcsin), we can solve for the two solutions:
For the first quadrant:
x₁ = arcsin(√(1/2))
For the second quadrant:
x₂ = π - arcsin(√(1/2))
Now, we can plug in the values of √(1/2) and evaluate arcsin:
x₁ = arcsin(√(1/2))
x₁ ≈ 0.7854 (in radians) or ≈ 45° (in degrees)
x₂ = π - arcsin(√(1/2))
x₂ ≈ 2.3562 (in radians) or ≈ 135° (in degrees)
So, the values of x in the interval [0, 2π] that satisfy the equation 6 sin^2(x) = 3 are approximately 0.7854 radians and 2.3562 radians (or approximately 45° and 135° in degrees).