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.