What is the exact value of cot(5(pi)/6)?

cot(5π/6)

= cos(5π/6) / sin(5π/6)
= sin(π/2 - 5π/6)/ cos(π/2 - 5π/6)
= sin(-π/3)/cos(-π/3)
= -sin(π/3)/cos(π/3)
= -(sqrt(3)/2)/(1/2)
= -sqrt(3)

Use Google, if you don't have a calculator. Type

tan(5*pi/6) =
in the search box.
Then take the reciprocal of that answer, for the cotangtent.
You may recognize the answer as the square root of 3.

5 pi/6 is a 150 degree angle. The reference angle is 30 degrees, but cosine is negative in the second quadrant.

To find the exact value of cot(5π/6), we need to recall the definitions and properties of the cotangent function and the values of trigonometric functions at special angles.

The cotangent of an angle θ is defined as the ratio of the adjacent side to the opposite side in a right triangle with angle θ. In terms of sine and cosine, the cotangent can be expressed as cot(θ) = cos(θ) / sin(θ).

To determine the exact value of cot(5π/6), we need to find the cosine and sine of 5π/6.

Step 1: Determine the cosine of 5π/6.

The cosine function represents the ratio of the adjacent side to the hypotenuse in a right triangle. In the case of 5π/6, it falls in the second quadrant, where the x-coordinate is negative. From the unit circle, we know that the cosine of 5π/6 is -(√3)/2.

Step 2: Determine the sine of 5π/6.

The sine function represents the ratio of the opposite side to the hypotenuse in a right triangle. In this case, the y-coordinate is positive. From the unit circle, we know that the sine of 5π/6 is 1/2.

Step 3: Calculate cot(5π/6).

Using the definition of cotangent, we can substitute the values of cosine and sine:

cot(5π/6) = cos(5π/6) / sin(5π/6) = (-(√3)/2) / (1/2) = -(√3)/2 * 2/1 = -√3.

Therefore, the exact value of cot(5π/6) is -√3.