Question

For what value of θ is cscθ undefined?

a. θ = 0
b. θ = π/2
c. θ = π
d. θ = 3π/2

Answer 1

The correct answers are options a and c. The csc function is undefined when the sine function is equal to zero. Therefore, cscθ is undefined for θ = 0 and θ = π.

Step-by-step explanation:

The csc (cosecant) function is defined as the reciprocal of the sine function, thus cscθ = 1/sinθ. For the csc function to be undefined, the sine function's value must be zero, as division by zero is undefined. Looking at the unit circle and the values of the sine function, we know that sine is equal to zero at 0 and π (or 180°), where the y-coordinate of the unit circle is zero. Therefore, for cscθ to be undefined, θ must be either 0 or π.

Between the options given in the question:

• Option a: θ = 0 - This makes sinθ = 0, and therefore it makes cscθ undefined.
• Option b: θ = π/2 - This makes sinθ = 1, and cscθ = 1, which is defined.
• Option c: θ = π - This also makes sinθ = 0, and hence cscθ would be undefined.
• Option d: θ = 3π/2 - This makes sinθ = -1, and cscθ would be -1, which is defined.

Based on the information above, the correct options are a. θ = 0 and c. θ = π.