Solve each equation on the interval [0, 2pi):
a. 4sin^2(x)-3=0
b. cos(3x)=-1
(2sinx-√3)(2sinx+√3) = 0
sinx = ±√3/2
x = π/3, 2π/3, 4π/3, 5π/3
cos(3x) = -1
so 3x = π,3π,5π
x = π/3, π, 5π/3
a. To solve the equation 4sin^2(x) - 3 = 0 on the interval [0, 2pi), follow these steps:
1. Move the constant term to the other side of the equation: 4sin^2(x) = 3.
2. Divide both sides of the equation by 4 to isolate sin^2(x): sin^2(x) = 3/4.
3. Take the square root of both sides of the equation: sin(x) = ±√(3/4).
4. Determine the possible values of sin(x) on the interval [0, 2pi). Since sin(x) is positive in the first and second quadrants, and we are looking for values within the interval [0, 2pi), the solution is limited to the first quadrant. Thus, sin(x) = √(3/4).
5. Find the values of x that satisfy sin(x) = √(3/4). Use the inverse sine function (sin^(-1)) to find the angle whose sine is √(3/4). This gives two solutions: x = sin^(-1)(√(3/4)) and x = π - sin^(-1)(√(3/4)).
6. Calculate the numerical values of x using a calculator:
- x = sin^(-1)(√(3/4)) ≈ 0.9553 radians ≈ 54.73 degrees
- x = π - sin^(-1)(√(3/4)) ≈ 2.1863 radians ≈ 125.27 degrees.
Therefore, the solutions to the equation 4sin^2(x) - 3 = 0 on the interval [0, 2pi) are approximately x = 0.9553 radians and x = 2.1863 radians (or approximately x = 54.73 degrees and x = 125.27 degrees).
b. To solve the equation cos(3x) = -1 on the interval [0, 2pi), follow these steps:
1. This equation involves the cosine function, which has a range of [-1, 1]. Since we are seeking a solution where cos(3x) = -1, we need to find when the cosine function equals its lower bound value of -1.
2. Determine the angle that gives a cosine of -1. This angle is π radians or 180 degrees.
3. Set the argument of the cosine function equal to this angle: 3x = π.
4. Divide both sides of the equation by 3: x = π/3.
5. Check if this solution lies within the given interval [0, 2pi). Since π/3 is approximately 1.0472 radians, which is within the interval [0, 2pi), the solution is valid.
Therefore, the solution to the equation cos(3x) = -1 on the interval [0, 2pi) is x = π/3 (or approximately x = 1.0472 radians).
a. To solve the equation 4sin^2(x) - 3 = 0 on the interval [0, 2pi), we can follow these steps:
Step 1: Move the constant term to the other side of the equation:
4sin^2(x) = 3
Step 2: Divide both sides of the equation by 4 to isolate sin^2(x):
sin^2(x) = 3/4
Step 3: Take the square root of both sides of the equation to solve for sin(x):
sin(x) = ±√(3/4)
Step 4: Determine the possible values of sin(x) on the interval [0, 2pi):
Since sin(x) is positive in the first and second quadrants, we have two cases to consider:
Case 1: sin(x) = √(3/4)
Using the inverse sine function, we find:
x = arcsin(√(3/4))
Since we are restricted to the interval [0, 2pi), x can take on the value of arcsin(√(3/4)).
Case 2: sin(x) = -√(3/4)
Using the inverse sine function, we find:
x = arcsin(-√(3/4))
Since we are restricted to the interval [0, 2pi), x can take on the value of arcsin(-√(3/4)).
Therefore, the solutions to the equation 4sin^2(x) - 3 = 0 on the interval [0, 2pi) are given by x = arcsin(√(3/4)) and x = arcsin(-√(3/4)).
b. To solve the equation cos(3x) = -1 on the interval [0, 2pi), we can follow these steps:
Step 1: We notice that -1 is the cosine value of π, so we can write:
3x = π + 2kπ (where k is an integer)
Step 2: Divide both sides of the equation by 3 to solve for x:
x = (π + 2kπ)/3
Step 3: Determine the values of k that satisfy the interval [0, 2pi):
Since the interval is [0, 2pi) and x is divided by 3, we need to find the values of k that make (π + 2kπ)/3 lie within this interval.
Considering k = 0:
x = (π + 2(0)π)/3 = π/3
Since π/3 is within the interval [0, 2pi), it is a valid solution.
Considering k = 1:
x = (π + 2(1)π)/3 = (π + 2π)/3 = (3π)/3 = π
Since π is within the interval [0, 2pi), it is a valid solution.
Considering k = 2:
x = (π + 2(2)π)/3 = (π + 4π)/3 = (5π)/3
Since (5π)/3 is outside the interval [0, 2pi), it is not a valid solution.
Therefore, the solutions to the equation cos(3x) = -1 on the interval [0, 2pi) are x = π/3 and x = π.