Problem 18 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Determine the Reference Angle

To find the reference angle for 210°, subtract 180° because 210° is in the third quadrant. The calculation is:\[ 210^{\circ} - 180^{\circ} = 30^{\circ} \]Thus, the reference angle is 30°.

Step 2: Recall the Cotangent of the Reference Angle

The cotangent of 30° can be derived using the relation:\[ \cot \theta = \frac{1}{\tan \theta} \] and \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \) which gives:\[ \cot 30^{\circ} = \sqrt{3} \]

Step 3: Determine the Sign in the Third Quadrant

In the third quadrant, both sine and cosine are negative, however, their ratio (cotangent) is positive. Therefore, the cotangent function is positive in the third quadrant.

Step 4: Combine Reference Angle and Sign Information

The reference angle for 210° is 30°, with a positive cotangent value. Therefore:\[ \cot 210^{\circ} = \cot 30^{\circ} = \sqrt{3} \]

Key Concepts

Cotangent

The cotangent is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function. Mathematically, it is expressed as \( \cot \theta = \frac{1}{\tan \theta} \). In other words, the cotangent of an angle is the ratio of the adjacent side to the opposite side in a right triangle.

One interesting aspect of the cotangent is its behavior across different quadrants in the unit circle. Depending on the quadrant, cotangent can either be positive or negative:

• In Quadrant I: Positive.
• In Quadrant II: Negative.
• In Quadrant III: Positive.
• In Quadrant IV: Negative.

In the given exercise, the angle 210° lies in the third quadrant, where cotangent is positive. Thus, finding the value of \( \cot 210^{\circ} \) involves recognizing these characteristics.

Reference Angle

A reference angle is a crucial concept in trigonometry. It is the smallest angle that the terminal side of a given angle makes with the horizontal axis. This provides a way to simplify and understand angles greater than 90° by relating them back to the first quadrant.

To find the reference angle for a given angle, follow these steps:

• For angles between 0° and 90°, the reference angle is the angle itself.
• For angles between 90° and 180°, subtract the angle from 180°.
• For angles between 180° and 270°, subtract 180° from the angle.
• For angles between 270° and 360°, subtract the angle from 360°.

In this specific example, 210° is in the third quadrant, and its reference angle is calculated as \( 210^{\circ} - 180^{\circ} = 30^{\circ} \). Knowing the reference angle allows you to find trigonometric values using known angles, like 30° in the unit circle.

Unit Circle

The unit circle is a powerful tool in trigonometry, offering a visual representation of angles and their corresponding trigonometric values. The circle has a radius of one unit centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle as well as the sine and cosine of that angle. This translates to all other trigonometric functions as they can be derived from sine and cosine.

Key aspects of the unit circle include:

• The right angle (90°) corresponds to points where coordinates are either (0,1) or (1,0).
• The special positions form angles like 30°, 45°, 60°, and their supplements.
• The coordinates (x, y) of any point on the unit circle correspond to (cos θ, sin θ).

For our given problem, utilizing the unit circle allows us to easily find common angle values like \( \cos 30^{\circ}, \sin 30^{\circ} \), and consequently, \( \tan 30^{\circ} \). This understanding aids in quick derivations, such as finding \( \cot 210^{\circ} \) by determining the reference angle's trigonometric values and adjusting for the quadrant's sign.