Problem 16 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\cot 210^{\circ}$$

Step-by-Step Solution

Answer

\(\cot 210^{\circ} = \sqrt{3}\).

Step 1: Identify the Quadrant

The angle \(210^{\circ}\) is in the third quadrant. Angles between \(180^{\circ}\) and \(270^{\circ}\) are in the third quadrant.

Step 2: Reference Angle Calculation

The reference angle for \(210^{\circ}\) is found by subtracting \(180^{\circ}\) from \(210^{\circ}\): \(210^{\circ} - 180^{\circ} = 30^{\circ}\). Therefore, the reference angle is \(30^{\circ}\).

Step 3: Determine the Sign of Cotangent

In the third quadrant, both sine and cosine are negative, hence \(\cot\theta = \frac{\cos\theta}{\sin\theta}\) is positive (as dividing two negative numbers results in a positive number).

Step 4: Calculate Cotangent of Reference Angle

For \(30^{\circ}\), the cotangent is \(\cot 30^{\circ} = \frac{\cos 30^{\circ}}{\sin 30^{\circ}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\).

Step 5: Apply the Sign to the Reference Angle Result

Since the cotangent is positive in the third quadrant and the reference angle result is \(\sqrt{3}\), the exact value of \(\cot(210^{\circ})\) is \(\sqrt{3}\).

Key Concepts

Cotangent

The cotangent is one of the six fundamental trigonometric functions. It is typically denoted as \( \cot \theta \), where \( \theta \) is the angle in question. Cotangent is essentially the reciprocal of the tangent function. Mathematically, it's expressed as:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• Or, equivalently, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This latter definition is essential because the sine and cosine functions are directly connected with the unit circle, making them foundational in determining other trigonometric values.
Understanding cotangent involves recognizing how it relates to the right triangle. In a right triangle, cotangent is the ratio of the adjacent side to the opposite side. However, it is most commonly used in various contexts like solving angles in circles and oscillations.

Reference Angle

A reference angle is a fundamental concept in trigonometry that aids in calculating the trigonometric functions of angles beyond the first quadrant. It is defined as the acute angle—less than or equal to \(90^{\circ}\)—formed by the terminal side of the angle and the horizontal axis.
For any given angle, the reference angle helps to evaluate its trigonometric functions by simplifying calculations using the corresponding acute angle in the first quadrant.

• For angles in the second quadrant: \( \text{Reference angle} = 180^{\circ} - \theta \)
• For the third quadrant: \( \text{Reference angle} = \theta - 180^{\circ} \)
• For the fourth quadrant: \( \text{Reference angle} = 360^{\circ} - \theta \)

It streamlines the process of finding the exact trigonometric values for various angles by reducing them to their simplest forms.

Trigonometric Quadrants

The concept of trigonometric quadrants involves dividing a circle into four sections, each 90 degrees. Starting from the positive x-axis and moving counterclockwise, the quadrants are numbered as follows: First Quadrant (0° to 90°), Second Quadrant (90° to 180°), Third Quadrant (180° to 270°), and Fourth Quadrant (270° to 360°).
Each quadrant has different signs for the sine, cosine, and tangent functions based on their coordinate values.

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine is positive, cosine and tangent are negative.
• Third Quadrant: Tangent is positive, sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, sine and tangent are negative.

Knowing the quadrant in which the angle lies helps determine the sign of the trigonometric function, which is crucial for evaluating trigonometric expressions.

Exact Values of Trigonometric Functions

Exact values for trigonometric functions are those that can be expressed in exact forms as opposed to decimal approximations. These are often derived from well-known angles such as 30°, 45°, and 60°, which occur frequently in mathematical problems and can be remembered or calculated easily.

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \)
• \( \cot 30^{\circ} = \sqrt{3} \)

These exact values are crucial in solving problems without resorting to a calculator. By knowing these, one can quickly infer the exact value of more complex trigonometric expressions like \( \cot(210^{\circ}) \) by using properties like cotangent and principles such as reference angles and quadrant sign rules.