Problem 27 - Algebra 2

Question

Evaluate each expression to the nearest hundredth. Each angle is given in radians. $$ \cot \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The cotangent of \( \frac{\pi}{6} \) to the nearest hundredth is 1.73.

Step 1: Define Cotangent

The first step is to understand what the cotangent function is. The cotangent of an angle in a right triangle is the ratio of the adjacent side to the opposite side (cotangent is the reciprocal of the tangent function).

Step 2: Use the Known Value

For the angle \( \frac{\pi}{6} \) or 30 degrees in degrees, the values of the trigonometric ratios are well known. Since the cotangent is the reciprocal of the tangent, and the tangent of 30 degrees is \( \frac{1}{\sqrt{3}} \), then the cotangent of 30 degrees or \( \frac{\pi}{6} \) is \( \sqrt{3} \).

Step 3: Round to the Nearest Hundredth

The square root of 3 is approximately 1.73 when rounded to the nearest hundredth.

Key Concepts

Cotangent

The cotangent function is one of the six fundamental trigonometric functions used in mathematics. It is denoted as 'cot'. Cotangent, specifically, is the reciprocal of the tangent function. This means that if you have the tangent of an angle (tan), then the cotangent is simply one divided by this value. In right triangle trigonometry, cotangent can be defined as the ratio of the length of the adjacent side to the length of the opposite side. This definition is particularly useful when dealing with right-angled triangles.
Understanding the cotangent function is pivotal when analyzing angle measures and their relationships. For instance, for an angle \( \theta \), if \( \tan(\theta) = \frac{1}{\sqrt{3}} \), then \( \cot(\theta) = \sqrt{3} \). This concept makes working with angles and their corresponding trigonometric ratios much more accessible.
In general, knowing the basic properties of cotangent helps in solving various trigonometric problems, such as finding angle measures or calculating side lengths in right triangles.

Radians

Radians are a unit of angular measurement that is widely used in mathematics, especially in trigonometry. Unlike degrees, which divide a circle into 360 parts, radians consider the radius of the circle.
The circumference of a circle is \( 2\pi \) times its radius. Therefore, one complete revolution around the circle corresponds to \( 2\pi \) radians. This makes \( \pi \) radians equal to 180 degrees.
When you are working with trigonometric functions, often angles are given in radians for ease of calculation, especially in calculus and other higher mathematics.

• To convert degrees into radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)
• To convert radians into degrees, use: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \)

Understanding radians can help simplify many mathematical problems and make connections with geometric interpretations more apparent.

Trigonometric Ratios

Trigonometric ratios are fundamentally important for understanding relationships in triangles, especially right triangles. The primary trigonometric ratios include sine, cosine, and tangent. Each of these ratios is a comparison between different sides of a right triangle relative to a particular angle.
For any angle \( \theta \):

• \( \sin(\theta) \) is the ratio of the opposite side to the hypotenuse.
• \( \cos(\theta) \) is the ratio of the adjacent side to the hypotenuse.
• \( \tan(\theta) \) is the ratio of the opposite side to the adjacent side.

These ratios can be used to solve for unknown sides or angles in right triangles, and they also extend into other areas of math, like calculus.
In addition to these primary ratios, we have the cotangent, secant, and cosecant, which are still based on the side-length ratios of our triangle.

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)

These extended ratios provide further insights and tools for exploring and solving trigonometric problems. Understanding and remembering these relationships is crucial for tackling trigonometry efficiently.