how is cot pi over 3 is root3?
The value of cot(pi/3) is equal to the reciprocal of the tangent of pi/3.
The tangent of pi/3 is √3 because in a unit circle, at an angle of pi/3, the y-coordinate of the point on the unit circle is 1/2, and the x-coordinate is √3/2. Therefore, the tangent is √3/1/2 = √3.
Since the cotangent is the reciprocal of the tangent, cot(pi/3) = 1/√3. Rationalizing the denominator by multiplying the numerator and denominator by √3, we get √3/√3*√3 = √3/3.
Therefore, cot(pi/3) is equal to √3/3 or root 3/3.
To find the value of cot(pi/3), we need to use the definition of the cotangent function. The cotangent of an angle is equal to the cosine of the angle divided by the sine of the angle.
Since we know that pi/3 is a special angle, we can use the unit circle to determine the values of sine and cosines for pi/3.
On the unit circle, the angle pi/3 corresponds to the point (1/2, sqrt(3)/2), where the x-coordinate represents the cosine and the y-coordinate represents the sine.
Therefore, cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2.
Now, we can substitute these values into the cotangent formula:
cot(pi/3) = cos(pi/3) / sin(pi/3)
= (1/2) / (sqrt(3)/2)
= 1 / sqrt(3)
To rationalize the denominator, we multiply both the numerator and the denominator by sqrt(3):
cot(pi/3) = (1 / sqrt(3)) * (sqrt(3) / sqrt(3))
= sqrt(3) / 3
Therefore, cot(pi/3) is equal to sqrt(3)/3.