Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \cot 30^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cot 30^{\circ} \) is \( \sqrt{3} \).

Step 1: Convert Degrees to Radians

The first step when dealing with trigonometric functions is often to convert the angle from degrees to radians. The conversion formula is given by: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}} \]Applying this to \(30^{\circ}\), we get: \[ 30^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{\pi}{6} \]

Step 2: Define Cotangent in terms of Sine and Cosine

The cotangent of an angle in trigonometry is defined as the reciprocal of the tangent. It can be expressed in terms of sine and cosine: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]For \( \theta = 30^{\circ} \), or \( \theta = \frac{\pi}{6} \), we'll need to find \( \cos \frac{\pi}{6} \) and \( \sin \frac{\pi}{6} \).

Step 3: Calculate Cosine and Sine of \(30^{\circ}\)

Remember the common values from the unit circle:- \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)- \( \sin 30^{\circ} = \frac{1}{2} \)

Step 4: Compute the Cotangent Value

Now that we have \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \), use the cotangent formula:\[ \cot 30^{\circ} = \frac{\cos 30^{\circ}}{\sin 30^{\circ}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} \]Simplify the expression:\[ \cot 30^{\circ} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3} \]

Key Concepts

Understanding Cotangent Value

The cotangent function, often seen as \( \cot \theta \), is one of the six fundamental trigonometric functions. It represents the reciprocal of the tangent function. Specifically, the cotangent of an angle \( \theta \) in a right triangle is the ratio of the adjacent side to the opposite side. In mathematical terms, it is written as:\[\cot \theta = \frac{\cos \theta}{\sin \theta}\]This means that to find the cotangent of an angle, you divide the cosine of that angle by the sine of the same angle. For instance, when \( \theta = 30^{\circ} \), the cotangent is determined by dividing the cosine value of \(30^{\circ}\) by its sine value.This relationship is useful because it allows trigonometric calculations to be performed even when a calculator does not provide direct cotangent values. By knowing the cosine and sine values from the unit circle, or memorizing common angle values, like \( \theta = 30^{\circ} \), it becomes easy to compute the cotangent directly.

Degree to Radian Conversion

Angles can be measured in degrees or radians. Converting between these units is essential in trigonometry since different contexts might require one unit over the other. The conversion formula itself is straightforward:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}}\]In this formula, \( \pi \approx 3.14159 \) represents the constant value that helps bridge the two systems of measurement. To convert \(30^{\circ}\) into radians, we multiply by \( \frac{\pi}{180^{\circ}} \), resulting in \( \frac{\pi}{6} \). This means that an angle of \(30^{\circ}\) corresponds to \( \frac{\pi}{6} \) radians.Converting angles is especially important when dealing with trigonometric functions in calculus and other advanced mathematics areas, where radians are the standard unit.

The Unit Circle and its Values

The unit circle is a circle with a radius of one that is centered at the origin of a coordinate plane. It provides a simple and effective way to remember the trigonometric function values of commonly used angles. Each point on the unit circle corresponds to an angle \( \theta \), measured from the positive x-axis, and helps easily find the sine and cosine values:

• Horizontal distance (x-coordinate) represents \( \cos \theta \)
• Vertical distance (y-coordinate) represents \( \sin \theta \)

For the angle \(30^{\circ}\) or \( \frac{\pi}{6} \), these values are \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \).These basic unit circle values simplify the calculations of trigonometric functions. Remembering these key angles can significantly speed up solving trigonometric problems, especially those involving cotangent, as these values directly feed into finding \( \cot 30^{\circ} = \sqrt{3} \).