Final answer:
To solve the equation csc x = -√2 for π ≤ x ≤ 3π/2, we need to find the angles in the third and fourth quadrants that satisfy sin x = -1/√2.
Step-by-step explanation:
The given equation is csc x = -√2 for π ≤ x ≤ 3π/2. To solve this equation, we need to find the angle x that satisfies the equation. First, we know that csc x is the reciprocal of sin x. So, we can write the equation as sin x = -1/√2. We know that the sine function is negative in the third and fourth quadrants, so we need to find the angle x in those quadrants that satisfies sin x = -1/√2.
In the third quadrant, the reference angle is π - x. So, we have sin(π - x) = -1/√2. Taking the reciprocal of both sides, we get csc(π - x) = -√2. Since π - x is between π and 3π/2, this satisfies the given equation.
In the fourth quadrant, the reference angle is x - 2π. So, we have sin(x - 2π) = -1/√2. Taking the reciprocal of both sides, we get csc(x - 2π) = -√2. Since x - 2π is between π and 3π/2, this satisfies the given equation.