What is the exact value of the cos of 23pi/12?

(Twenty-three pi over 12, which is = to 345 degrees.)

Thank you

To find the exact value of cos(23π/12), we can use the unit circle.

First, let's convert 23π/12 to degrees. We know that 360° is equal to 2π radians. Therefore, to convert radians to degrees, we can multiply by (180°/π).

23π/12 * (180°/π) = 23 * 15°/2 = 345°

Now, we want to find the cosine of 345 degrees.

Remember that the unit circle represents the cosine values for different angles. For 345 degrees, we first need to determine the equivalent reference angle in the range of 0 to 360 degrees.

Since 345 degrees is in the fourth quadrant (between 270 and 360 degrees), the reference angle will be 360 - 345 = 15 degrees.

On the unit circle, the cosine of any angle is represented by the x-coordinate of the point on the circle corresponding to that angle.

For the reference angle of 15 degrees, the cosine value can be found by locating the point on the unit circle that is 15 degrees counter-clockwise from the positive x-axis.

The point on the unit circle for 15 degrees has coordinates (cos(15°), sin(15°)), where cos(15°) is the cosine value we are looking for.

Using trigonometric ratios, we know that sin(15°) = sin(180° - 15°) = sin(165°), and since the unit circle is symmetrical about the x-axis, sin(165°) = -sin(15°). Therefore, the y-coordinate of our point is -sin(15°).

Now, we can use the Pythagorean identity to find the value of sin(15°):
sin²(15°) + cos²(15°) = 1

Since sin(15°) = -sin(15°), we have:
(-sin(15°))² + cos²(15°) = 1
sin²(15°) + cos²(15°) = 1
cos²(15°) = 1 - sin²(15°)
cos(15°) = sqrt(1 - sin²(15°))

We can calculate the value of sin(15°) by using a half-angle formula:
sin(15°) = sqrt((1 - cos(30°))/2)

Given that cos(30°) = sqrt(3)/2, we substitute it into the equation:
sin(15°) = sqrt((1 - sqrt(3)/2))/2

Now, we can substitute the value of sin(15°) into the equation for cos(15°):
cos(15°) = sqrt(1 - (sqrt(1 - sqrt(3)/2)/2)²)

Simplifying this expression will give us the exact value of cos(15°).