Answer:
To solve the equation cot(x - π/2) = 1 on the interval [0, 2π), we can follow these steps:
Step 1: Simplify the equation
We know that cot(x - π/2) is equal to the reciprocal of tan(x - π/2). So, we can rewrite the equation as:
tan(x - π/2) = 1
Step 2: Find the reference angle
Since the tangent function repeats every π radians, we can find the reference angle by subtracting π/2 from x and finding an equivalent angle within the interval [0, π).
So, x - π/2 = reference angle
Step 3: Solve for the reference angle
We need to find the angle whose tangent is equal to 1. In the first quadrant, the tangent of an angle is positive, and the reference angle is π/4.
So, the reference angle is π/4.
Step 4: Find the solutions within the given interval
Since the tangent function has a period of π, we can add multiples of π to the reference angle to find all the solutions within the interval [0, 2π).
Adding π, we get:
π/4 + π = 5π/4
Adding another π, we get:
π/4 + 2π = 9π/4
So, the solutions within the interval [0, 2π) are x = 5π/4 and x = 9π/4.
In summary, the solutions to the equation cot(x - π/2) = 1 on the interval [0, 2π) are x = 5π/4 and x = 9π/4.
Explanation:
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