Problem 31 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \cot 270^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cot 270^{\circ} \) is 0.

Step 1: Understanding Cotangent

The cotangent function, \( \cot \theta \), is defined as the reciprocal of the tangent function, \( \tan \theta \). Therefore, \( \cot \theta = \frac{1}{\tan \theta} \). In other words, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Step 2: Analyzing a 270° Angle on the Unit Circle

On the unit circle, 270° is the angle located on the negative \( y \)-axis. At this angle, the coordinates are \((0, -1)\), which means \( \cos 270^{\circ} = 0 \) and \( \sin 270^{\circ} = -1 \).

Step 3: Applying the Cotangent Definition

Using the definition of cotangent, \( \cot 270^{\circ} = \frac{\cos 270^{\circ}}{\sin 270^{\circ}} = \frac{0}{-1} \).

Step 4: Simplifying the Expression

The expression \( \frac{0}{-1} \) simplifies to 0. Therefore, \( \cot 270^{\circ} = 0 \).

Key Concepts

Unit Circle

A very useful tool in trigonometry is the unit circle, primarily because it helps visualize trigonometric functions based on angle rotations. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. This is the unit circle. Each point on this circle corresponds to a particular angle in standard position, measured in a counterclockwise direction from the positive x-axis.
Trigonometric ratios like sine, cosine, and cotangent can be directly related to the coordinates of these points:

• The x-coordinate gives the cosine of the angle.
• The y-coordinate provides the sine of the angle.

Thus, the point (0, -1) represents the angle of 270° on the unit circle. Here, \( \cos 270^{\circ} = 0 \) and \( \sin 270^{\circ} = -1 \). Recognizing the coordinates becomes crucial because these values are directly used in computing trigonometric functions, such as cotangent.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of a triangle, primarily used in the study of periodic phenomena and waves. These include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), tangent (\( \tan \theta \)), and cotangent (\( \cot \theta \)), among others.
To understand cotangent, consider its relationship to other trigonometric functions:

• It is the reciprocal of tangent:\( \cot \theta = \frac{1}{\tan \theta} \)
• \( \cot \theta \) can also be expressed in terms of sine and cosine:\( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This is particularly useful when you know the values of sine and cosine for a given angle, allowing you to swiftly compute cotangent. The cotangent function is periodic, repeating every 180° or \( \pi \) radians.

Angle Measurement

Understanding angle measurement is essential for working with trigonometric functions on the unit circle. Angles can be measured in degrees or radians. In the unit circle:

• 360° corresponds to a complete revolution or 2π radians.
• An angle of 270°, like in our example, represents three-quarters of a turn around the circle. In radians, this is\( \frac{3\pi}{2} \)

For practical calculations, degrees are often easier to visualize, but radians are more beneficial in higher mathematics. In our problem, knowing that 270° places us directly on the negative y-axis helped determine the cosine and sine values for the cotangent calculation. This demonstrates how important understanding angle measurement is in manipulating and analyzing functions related to angles.