State the exact value for 11π/6 Show all steps

Question 4 (4 points) State the exact value for

. Show all steps. Include a diagram as part of your solution

The Correct Answer and Explanation is:

Of course. Here is the solution with all the steps and a diagram as requested.

Final Answer:
The exact value of cot(11π/6) is -√3.

Step-by-Step Solution

Step 1: Locate the Angle

First, we need to determine the position of the angle 11π/6 on the unit circle. An angle is measured counterclockwise from the positive x-axis. We know that a full circle is 2π radians.

To make it easier to locate, we can express 2π with a denominator of 6:
2π = 12π/6

The angle 11π/6 is slightly less than 12π/6. This means the terminal side of the angle 11π/6 lies in the fourth quadrant (Quadrant IV).

Step 2: Find the Reference Angle

The reference angle is the acute angle that the terminal side of 11π/6 makes with the x-axis. Since the angle is in Quadrant IV, we can find the reference angle (α) by subtracting 11π/6 from the full circle (2π).

Reference Angle (α) = 2π – 11π/6
α = 12π/6 – 11π/6
α = π/6

The reference angle is π/6. This means the trigonometric values of 11π/6 will have the same magnitude as the values for π/6, but the sign will be determined by the quadrant.

Step 3: Determine the Sign of Cotangent in Quadrant IV

The cotangent function is defined as cot(θ) = cos(θ) / sin(θ). On the unit circle, this corresponds to the ratio of the x-coordinate to the y-coordinate (x/y).

In Quadrant IV, the x-coordinate is positive.
In Quadrant IV, the y-coordinate is negative.

Therefore, cotangent is negative in Quadrant IV because (positive) / (negative) = negative.

Step 4: Calculate the Exact Value

Now, we calculate the value of cotangent for the reference angle, π/6.

cot(π/6) = cos(π/6) / sin(π/6)

We know the standard values for π/6:

cos(π/6) = √3 / 2
sin(π/6) = 1/2

So, cot(π/6) = (√3 / 2) / (1/2) = √3.

Finally, we apply the negative sign we determined in Step 3.

cot(11π/6) = -cot(π/6) = -√3

Diagram

The following diagram illustrates the angle 11π/6 in standard position on the unit circle.Generated code

y-axis
        ^
        |
        |
  QII   |   QI
        |
--------|--------> x-axis
        |
  QIII  |  QIV
        |     /
        |    / α = π/6
        |   /
        |  /  <-- Terminal side of 11π/6
        * (√3/2, -1/2)

Diagram Explanation:
The diagram shows the angle 11π/6 starting from the positive x-axis and rotating counterclockwise, ending in the fourth quadrant. The reference angle, α = π/6, is the acute angle between the terminal side and the positive x-axis. The point where the terminal side intersects the unit circle has coordinates (cos(11π/6), sin(11π/6)), which are (√3/2, -1/2). The cotangent is the x-coordinate divided by the y-coordinate, confirming the result: (√3/2) / (-1/2) = -√3

Step-by-Step Solution

Diagram

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