Final answer:
The cosecant function csc θ is undefined for θ = nπ radians (0°, 180°, 360°, etc.), where n is an integer, because that is when the sine function sin θ is zero.
Step-by-step explanation:
The value of θ for which csc θ is undefined corresponds to the angles where the sine function, sin θ, equals zero. The cosecant function, csc θ, is the reciprocal of the sine function, csc θ = 1/sin θ. Therefore, csc θ is undefined when sin θ = 0.
The sine function is zero at angles of nπ, where n is an integer (0, ± 1, ± 2, ...), which means at 0°, 180°, 360°, etc. So, the value of θ that makes csc θ undefined is when θ = nπ radians or equivalent angles in degrees (0°, 180°, 360°, etc.).
The value of csc θ is undefined when the sine of θ is equal to zero. Since csc θ = 1/sin θ, when the sine of θ is zero, the value of csc θ becomes undefined because division by zero is undefined in mathematics.
The sine of an angle θ is equal to zero when the angle is a multiple of 180 degrees, or θ = 180n, where n is an integer. So, for any value of θ = 180n, the value of csc θ is undefined.