cot (pi/12)
pi / 12 = 15 °
pi / 12 = 30 ° / 2
sin 30 ° = 1 / 2
cos 30 ° = sqrt ( 3 ) / 2
cot ( theta / 2 ) = ( 1 + cos theta ) / sin theta
cot 15 ° = ( 1 + cos 30 ° ) / sin 30 °
cot 15 ° = [ 1 + sqrt ( 3 ) / 2 ] / ( 1 / 2 )
cot 15 ° = 2 * [ 1 + sqrt ( 3 ) / 2 ]
cot 15 ° = 2 * [ 2 / 2 + sqrt ( 3 ) / 2 ) ]
cot 15 ° = 2 * [ 2 + sqrt ( 3 ) ] / 2
cot 15 ° = 2 + sqrt ( 3 )
cot pi / 12 = cot 15 ° = 2 + sqrt ( 3 )
To find the value of cot(pi/12), we can use the formula for cotangent.
The cotangent function (cot) of an angle is the ratio of the adjacent side to the opposite side of a right triangle.
First, we need to identify the right triangle where the angle pi/12 is located. To do this, imagine a coordinate plane with angles starting from the positive x-axis in a counter-clockwise direction.
For pi/12, we need to find a triangle where the reference angle is pi/12. The reference angle is the acute angle between the terminal side of the given angle and the x-axis.
Since pi/12 is less than pi/4 (45 degrees), we know that the reference angle is pi/12.
Now, let's draw a right triangle with the reference angle pi/12 in the coordinate plane.
The adjacent side is the one adjacent to the angle, and the opposite side is the one opposite to the angle.
In this case, the adjacent side is the side adjacent to the reference angle, and the opposite side is the side opposite to the reference angle.
We need to determine the lengths of these sides. To do this, we can use the triangle's relationships based on a unit circle.
Imagine a unit circle centered at the origin (0,0) with a radius of 1 unit. The angle pi/12 is on the unit circle.
The x-coordinate of the point where the angle intersects the unit circle represents the adjacent side, and the y-coordinate represents the opposite side.
Using the values on the unit circle, we can find the x and y coordinates of this point.
For the angle pi/12, the coordinates of the point on the unit circle are:
x = cos(pi/12)
y = sin(pi/12)
Now, we can calculate these values using a scientific calculator or trigonometric identities.
cos(pi/12) ≈ 0.9659
sin(pi/12) ≈ 0.2588
Now, we know that the adjacent side is approximately 0.9659 units long and the opposite side is approximately 0.2588 units long.
Finally, we can use the formula for cotangent to find the value of cot(pi/12):
cot(pi/12) = adjacent side / opposite side
cot(pi/12) = (approx. 0.9659) / (approx. 0.2588)
Evaluating this expression, we get:
cot(pi/12) ≈ 3.7321
Therefore, cot(pi/12) is approximately 3.7321.