Find the EXACT VALUE of csc(-11pi/12)

If that angle is in radians, it is easy.

A full circle has 2PI radians...or 360 deg

so 22PI/24 is the same as 22*15 deg, or -30 deg. The 30 deg triangle is well known. The sin is 1/2, so the csc is 2

Figure out the sign of the csc from what quadrant it is in.

To find the exact value of csc(-11π/12), we need to determine the reference angle and quadrant in which the angle lies.

First, let's convert -11π/12 into degrees. As you mentioned, a full circle is 2π radians or 360 degrees. Therefore, we can use the conversion ratio:

-11π/12 * (180 degrees/π radians) = -165 degrees.

The angle is -165 degrees. Now, we need to find the reference angle in the range 0 to 90 degrees.

To do this, we can add 360 degrees until we get a positive angle:

-165 degrees + 360 degrees = 195 degrees.

The reference angle for -165 degrees is 195 degrees.

Next, we determine the sign of csc based on the quadrant. Recall that:

- In the first quadrant, all trigonometric ratios are positive.
- In the second quadrant, only sin and csc are positive.
- In the third quadrant, only tan and cot are positive.
- In the fourth quadrant, only cos and sec are positive.

Since the reference angle of 195 degrees lies in the third quadrant, only tan and cot are positive. Therefore, csc(-11π/12) is negative.

Now, let's find the exact value of csc(-11π/12) using the reference angle:

Since the reference angle is 195 degrees, we can use the well-known 30-60-90 right triangle with a reference angle of 30 degrees. In this triangle, the sin of 30 degrees is 1/2.

However, because csc is the reciprocal of sin, the csc of 30 degrees is the inverse of 1/2, which is 2.

Since csc(-11π/12) has the same magnitude as csc(30 degrees) but with a negative sign, the exact value of csc(-11π/12) is -2.

Therefore, csc(-11π/12) = -2.