Understanding cotangent (cot) and finding the value of cot(π/3) using the unit circle and special triangle

cot(π/3)

To find the value of cot(π/3), we need to understand the concept of the cotangent function

To find the value of cot(π/3), we need to understand the concept of the cotangent function.

The cotangent (cot) of an angle is the ratio of the adjacent side to the opposite side in a right triangle. It is the reciprocal of the tangent function.

In the given problem, we are asked to find the value of cot(π/3).

To evaluate cot(π/3), we can use the unit circle.

The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) in a coordinate plane. The circumference of the unit circle is 2π.

To find the value of cot(π/3), we need to identify the point on the unit circle that corresponds to the angle of π/3.

When an angle θ is in standard position (vertex at the origin, initial side along the positive x-axis), we can determine the coordinates of the point P on the unit circle by setting x = cos(θ) and y = sin(θ).

For the angle π/3, its coordinates on the unit circle are (cos(π/3),sin(π/3)).

Using the special triangle for π/3, which is an equilateral triangle with all sides equal to 2, we can find that cos(π/3) = 1/2 and sin(π/3) = √3/2.

Therefore, the coordinates of the point corresponding to π/3 on the unit circle are (1/2, √3/2).

Now, cot(θ) is defined as the ratio of the adjacent side to the opposite side in a right triangle.

Since we are dealing with the point on the unit circle, the adjacent side is the x-coordinate and the opposite side is the y-coordinate.

Therefore, cot(π/3) = adjacent/opposite = x/y = (1/2) / (√3/2).

To simplify this expression, multiply the numerator and denominator by 2/√3:

cot(π/3) = (1/2) / (√3/2) * (2/√3) = 1 / √3.

To rationalize the denominator, we multiply the numerator and denominator by √3:

cot(π/3) = 1 / √3 * √3/√3 = √3/3.

Thus, cot(π/3) is equal to √3/3.

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