2cotx + 1 = -1 over [0,2pi)
2cotx = -2
cotx = -1
as a reference angle, you know cot pi/4 = 1, so since cotx is negative in QII and QIV,
x = 3pi/4, 7pi/4
Ma ogi nh
To solve the equation 2cot(x) + 1 = -1 over the interval [0, 2π), we can follow these steps:
1. Move the constant term to the other side:
2cot(x) = -2 - 1
Simplify:
2cot(x) = -3
2. Divide both sides of the equation by 2:
cot(x) = -3/2
3. To find the values of x that satisfy this equation over the interval [0, 2π), we can use the fact that cot(x) is the reciprocal of tan(x). Therefore, the equation cot(x) = -3/2 is equivalent to tan(x) = -2/3.
4. Use the inverse tangent function (usually denoted as arctan or atan) to find the angle whose tangent is -2/3. We can do this by taking the arctan of both sides of the equation:
x = arctan(-2/3)
5. Calculate the principal solution in radians by applying the arctan function:
x ≈ -0.588
6. However, we need to consider the interval [0, 2π). Since the arctan function only returns angles within the range (-π/2, π/2), we need to adjust the angle to fall within the given interval.
By adding 2π to the angle, we can find another angle that satisfies the equation within the interval. Therefore:
x ≈ -0.588 + 2π ≈ 5.694
So, the solutions to the equation 2cot(x) + 1 = -1 over [0, 2π) are approximately x ≈ -0.588 and x ≈ 5.694.