What is the exact value of cot(5pi/6)?
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To find the exact value of cot(5π/6), we need to understand the concept of the cotangent function and use trigonometric identities.
1. Recall that the cotangent function (cotθ) is defined as the ratio of the adjacent side to the opposite side in a right triangle. It can also be expressed as the reciprocal of the tangent function: cotθ = 1/tanθ.
2. In this case, the angle is 5π/6. Visualize a right triangle with this angle. Since the angle is greater than π/2 (90 degrees) and less than π (180 degrees), it lies in the second quadrant.
3. In the second quadrant, the adjacent side is negative, and the opposite side is positive.
4. To find the exact value of cot(5π/6), we can use the relationship between the tangent and cotangent functions. We recall that tanθ = sinθ / cosθ. Using this, we can find the tangent of 5π/6 as follows:
tan(5π/6) = sin(5π/6) / cos(5π/6)
5. The sine function (sinθ) is positive in the second quadrant, while the cosine function (cosθ) is negative. Therefore, we have:
tan(5π/6) = sin(5π/6) / (-cos(5π/6))
6. The exact value of sin(5π/6) can be determined from the unit circle or trigonometric table. We know that sin(π/6) = 1/2, and taking the negative value for the second quadrant:
sin(5π/6) = -1/2
7. Similarly, the exact value of cos(5π/6) is known from the unit circle or trigonometric table. We know that cos(π/6) = √3/2, and again taking the negative value:
cos(5π/6) = -(√3/2)
8. Plugging these values into the tangent equation from step 5:
tan(5π/6) = (-1/2) / (-(√3/2))
9. Simplifying the expression:
tan(5π/6) = 1 / √3
10. Since cot(θ) = 1/tan(θ), we find the value of cot(5π/6) as:
cot(5π/6) = 1 / (1 / √3)
11. Multiplying the numerator and denominator by √3 (to rationalize the denominator):
cot(5π/6) = √3 / 1
Simplifying the expression:
cot(5π/6) = √3
So, the exact value of cot(5π/6) is √3.