Question

Find the exact value of each expression. Do not use a calculator. $$ \cot \left(-\frac{\pi}{6}\right) $$

Short answer

-\sqrt{3}

Step-by-step solution

Understand the Cotangent Function

The cotangent function is defined as the reciprocal of the tangent function: \[\cot(\theta) = \frac{1}{\tan(\theta)}.\]

Use the Even-Odd Properties of Trigonometric Functions

Recognize that cotangent is an odd function. This means \(\cot(-\theta) = -\cot(\theta)\), so we can rewrite the expression: \[\cot\left(-\frac{\pi}{6}\right) = -\cot\left(\frac{\pi}{6}\right).\]

Determine the Value of \(\cot\left(\frac{\pi}{6}\right)\)

We know from the properties of special angles that: \[\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}.\] Therefore, \[\cot\left(\frac{\pi}{6}\right) = \frac{1}{\tan\left(\frac{\pi}{6}\right)} = \sqrt{3}.\]

Apply the Even-Odd Identity

Using Step 2, we now have: \[\cot\left(-\frac{\pi}{6}\right) = -\sqrt{3}.\]

Final Value

The exact value of \(\cot\left(-\frac{\pi}{6}\right)\) is \(-\sqrt{3}\).

Key concepts

Reciprocal Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent all have reciprocal counterparts. The reciprocal of the tangent function is the cotangent function. This means that \(\text{cot}(\theta) = \frac{1}{\text{tan}(\theta)}\). Learning the relationships between a function and its reciprocal is crucial in trigonometry. If you know the value of one, you can easily find the value of the other. For instance, if \(\text{tan}(\theta)\) is known, you can calculate \(\text{cot}(\theta)\) quickly by taking the reciprocal. Understanding these reciprocal relationships helps in simplifying complex trigonometric expressions or solving equations.

Odd-Even Properties

Trigonometric functions exhibit unique behavior when dealing with negative inputs. These behaviors are captured by the concepts of odd and even functions. An odd function has the property that \(-f(x) = f(-x)\), while an even function satisfies \(-f(x) = -f(x)\). Cotangent is an example of an odd function, so \(\text{cot}(-\theta) = -\text{cot}(\theta)\). Recognizing these properties simplifies problem-solving. For instance, in the original exercise, we used the fact that \(\text{cot}(-\frac{\text{π}}{6}) = -\text{cot}(\frac{\text{π}}{6})\), which immediately directed us to the correct value.

Special Angles in Trigonometry

Certain angles, often referred to as 'special angles,' have well-known trigonometric values. This includes angles like \(\frac{\text{π}}{6}\), \(\frac{\text{π}}{4}\), and \(\frac{\text{π}}{3}\). These angles are frequently encountered in trigonometric problems. For example, it is known that \(\text{tan}(\frac{\text{π}}{6}) = \frac{1}{\text{√3}}\), which means \(\text{cot}(\frac{\text{π}}{6}) = \text{√3}\). Memorizing these values can save time and simplify calculations. The special angle values are fundamental because they serve as building blocks for understanding more complex trigonometric functions, identities, and equations.