Question

Find the value of the trigonometric function. If possible, give the exact value; otherwise, use a calculator to find an approximate value correct to five decimal places. csc 2π

Answer 1

The value of csc 2π is undefined because it is the reciprocal of sin 2π, which is 0, and the reciprocal of zero is not defined.

Step-by-step explanation:

To find the value of csc 2π, recall that csc is the reciprocal of the sine function. Since sin(2π) equals 0 (as 2π corresponds to one complete cycle on the unit circle, bringing us back to the starting point on the positive x-axis), csc(2π) would be the reciprocal of 0. However, the reciprocal of zero is undefined, so csc 2π does not have a value and is undefined.

For trigonometric functions like sine and cosine, the values at key angles such as 0, π/2, π, 3π/2, and 2π are typically known from the unit circle. Specifically, sin(2π) = 0, cos(2π) = 1, and thus csc(2π), being 1/sin(2π), cannot be calculated and is undefined.