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Chapter 6: Problem 12

Find the exact circular function value for each of the following. $$\cot \frac{5 \pi}{6}$$

Short Answer

Expert verified
The exact value for \(\text{cot} \frac{5 \pi}{6} \) is \(-\sqrt{3}\).

Step by step solution

01

Identify the Angle in Radians

Recognize that the given angle is \(\frac{5 \pi}{6}\). This angle can be placed in the unit circle for reference.
02

Locate the Angle in the Unit Circle

The angle \(\frac{5 \pi}{6}\) lies in the second quadrant because \(\frac{5 \pi}{6} \) is slightly less than \(\pi\) radians (180 degrees). This places it in the second quadrant.
03

Determine Reference Angle and Value

Find the reference angle for \(\frac{5 \pi}{6}\), which is \(\pi - \frac{5 \pi}{6} = \frac{\pi}{6}\). We now use the reference angle to find the required trigonometric values.
04

Use Properties of Cotangent

The cotangent function is defined as \( \text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)} \).
05

Substitute Known Values

From the unit circle, we know \(\text{sin}\frac{\pi}{6} = \frac{1}{2}\) and \(\text{cos}\frac{\pi}{6} = \frac{\sqrt{3}}{2}\). Since we are in the second quadrant and cosine is negative while sine is positive: \( \text{cos} \frac{5 \pi}{6} = -\frac{\sqrt{3}}{2}, \text{sin} \frac{5 \pi}{6} = \frac{1}{2} \).
06

Calculate Cotangent Value

Substitute the known values for cosine and sine into the cotangent function: \( \text{cot} \frac{5 \pi}{6} = \frac{ \text{cos} \frac{5 \pi}{6} }{ \text{sin} \frac{5 \pi}{6} } = \frac{ -\frac{\sqrt{3}}{2} }{ \frac{1}{2} } = -\sqrt{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
The unit circle is a powerful tool in trigonometry. It's a circle with a radius of 1 centered at the origin on the coordinate plane. Every point on this circle corresponds to an angle's cosine and sine values.
The angle in question, \( \frac{5 \pi}{6} \), can be plotted on the unit circle.

Here are some essential facts about the unit circle:
  • The coordinates of any point on it are (cos 胃, sin 胃).

  • The angle 胃 (in radians) is measured from the positive x-axis.

  • Angles in the unit circle can go from 0 to 2蟺 for one full revolution.
Knowing these basics helps in determining the trigonometric functions for specific angles quickly and accurately.
Cotangent
Cotangent, often abbreviated as 'cot,' is one of the six main trigonometric functions.
The cotangent of an angle 胃 is defined as the reciprocal of the tangent of 胃. In mathematical terms:
\[ \text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)} \]

This means that cotangent is the ratio of the adjacent side to the opposite side in a right-angled triangle when the angle is 胃.
  • Cotangent is positive in the first and third quadrants.

  • It is negative in the second and fourth quadrants.
Understanding this concept helps in solving problems where you need to find the cotangent of particular angles.
Reference Angle
Reference angles are smaller angles that help in calculating trigonometric functions for larger angles.
To find a reference angle, you essentially 'fold' a larger angle back into the first quadrant. The reference angle will always be between 0 and \(\frac{\pi}{2} \) or 0 and 90 degrees.
Here's how you find the reference angle for \( \frac{5 \pi}{6} \):
  • It's in the second quadrant.

  • Subtract the angle from \( \pi \): \[ \pi - \frac{5 \pi}{6} = \frac{ \pi}{6} \]
This simplifies our problem, making it easier to use known sine and cosine values of these simple reference angles.
Radian Measure
Radians are another way to measure angles, besides degrees.
A radian measures the angle created when the radius of a circle is wrapped along its circumference.
One complete revolution around the unit circle is 2蟺 radians, which is equivalent to 360 degrees:
  • \( \pi \) radians is equal to 180 degrees.

  • \( \frac{\pi}{2} \) radians is equal to 90 degrees.
In our case, \( \frac{5 \pi}{6} \) radians is just under 蟺 radians (or 180 degrees), positioning it in the second quadrant. This understanding is crucial for accurately plotting and interpreting angles on the unit circle.

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