Evaluate giving exact values and identities. Calculator based solutions are not acceptable. Thanks for any help

sin (-pi/12)

This has been answered elsewhere. I liked Reiny's suggestion that you evaluate

sin (30 degrees - 45 degrees),
or, if you prefer,
sin (pi/6 - pi/4)
using the formula for sin (A-B)

To determine the value of sin(-pi/12) without using a calculator, we can follow these steps:

Step 1: Recognize the reference angle.
The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis. In this case, the reference angle is pi/12 because it is the positive angle obtained by rotating -pi/12 counterclockwise to the nearest x-axis.

Step 2: Identify the quadrant.
Since -pi/12 is negative, it falls in either the third or fourth quadrant. To determine the correct quadrant, we consider the sign of sin.
In the third quadrant (Q3), sin is positive. However, in the fourth quadrant (Q4), sin is negative. So, sin(-pi/12) will be negative.

Step 3: Determine the exact value using the reference angle.
We can use the following trigonometric identity:
sin(-a) = -sin(a)
This identity tells us that the sine of a negative angle is equal to the negative sine of the positive angle.

Since the reference angle is pi/12, we need to find the sine of pi/12 and then negate it.
Unfortunately, there is no exact numerical value for sin(pi/12). However, we can express it using trigonometric identities.

One identity that can help us is the half-angle identity for sine:
sin(a/2) = sqrt((1 - cos(a)) / 2)

Using this identity, we can find sin(pi/6), which is equivalent to sin(pi/12):

sin(pi/6) = sqrt((1 - cos(pi/6)) / 2)
= sqrt((1 - sqrt(3)/2) / 2)
= sqrt((2 - sqrt(3)) / 4)

Now, we substitute sin(pi/6) back into our original equation:

sin(-pi/12) = -sin(pi/12)
= -sin(pi/6)
= -sqrt((2 - sqrt(3)) / 4)

Therefore, the exact value of sin(-pi/12) is -sqrt((2 - sqrt(3)) / 4).