What is the exact value of cot(4pi/3)?

Do not use a calculator.

4pi/3 is in quadrant III

the tangent of a 3rd quadrant angle is positive so
cot(4pi/3)
= 1/tan(4pi/3) , the angle in standard position is pi/3
= 1/tanpi/3
= 1/√3

To find the exact value of cot(4π/3), we need to understand the values of cosine and sine for this angle.

First, let's find the value of cos(4π/3):
The angle 4π/3 is in the third quadrant of the unit circle. In the third quadrant, the x-coordinate is negative, and the y-coordinate is negative.

In the unit circle, we know that cos(θ) = x-coordinate and sin(θ) = y-coordinate.

For 4π/3, the x-coordinate is -1/2, and the y-coordinate is -√3/2.

Now, let's find cot(4π/3):
cot(θ) = cos(θ) / sin(θ)

Using the values we found earlier, we plug them into the cot formula:
cot(4π/3) = (-1/2) / (-√3/2)

To simplify this expression, we multiply the numerator and denominator by the reciprocal of the denominator:

cot(4π/3) = (-1/2) * (-2/√3)

Simplifying further gives:
cot(4π/3) = 1 / √3

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

cot(4π/3) = (1/√3) * (√3/√3)

Finally, we get the exact value of cot(4π/3) as:
cot(4π/3) = √3 / 3

To find the exact value of cot(4π/3), we can use the relationship between cotangent and tangent.

The relationship between cotangent (cot) and tangent (tan) is:

cot(x) = 1 / tan(x)

So, to find cot(4π/3), we need to find the value of tan(4π/3) first.

The angle 4π/3 is in the third quadrant of the unit circle.

In the unit circle, the tangent function is positive in the first and third quadrants.

To find tan(4π/3), we can find the tangent of the reference angle, which is π/3.

The reference angle is the angle between the terminal side of the angle and the x-axis.

In the third quadrant, the reference angle is π - π/3, which simplifies to 2π/3.

The tangent function of 2π/3 can be found as follows:

tan(2π/3) = sin(2π/3) / cos(2π/3)

Using the unit circle values, we can find that:

sin(2π/3) = √3/2
cos(2π/3) = -1/2

Substituting these values into the equation for tan(2π/3), we get:

tan(2π/3) = (√3/2) / (-1/2)
= -(√3/2) * (2/-1)
= -(√3/2) * (-2/1)
= (√3/1)
= √3

Now that we have the value of tan(4π/3) as √3, we can find cot(4π/3) using the relationship between cotangent and tangent:

cot(4π/3) = 1 / tan(4π/3)

Substituting the value of tan(4π/3) = √3, we get:

cot(4π/3) = 1 / √3

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

cot(4π/3) = (1 / √3) * (√3 / √3)
= √3 / 3

Therefore, the exact value of cot(4π/3) is √3 / 3.