Problem 21 - Algebra 2

Question

Evaluate each expression to the nearest hundredth. Each angle is given in radians. $$ \cot 3 $$

Step-by-Step Solution

Answer

\(-0.1425\)

Step 1: Determining the Cotangent Function

The cotangent function, represented as \( \cot \), is the reciprocal of the tangent function. This means that \( \cot x = \frac{1}{\tan x} \).

Step 2: Evaluating the Tangent of the Given Angle

The next step is to evaluate the tangent of the given angle. Make sure your calculator is set to radian mode (not degree mode) because our input angle is given in radians. Then calculate \( \tan 3 \).

Step 3: Calculating the Cotangent

After obtaining the value of \( \tan 3 \), you calculate the cotangent by taking its reciprocal. This means you will do \( \frac{1}{\tan 3} \) to find the cotangent of 3.

Key Concepts

Cotangent Function

The cotangent function is an essential trigonometric function that ties closely with the tangent function. Unlike the tangent function, which is defined as the ratio of the sine to the cosine of an angle, the cotangent function is its reciprocal.

In mathematical terms, you can express this relationship as:

• \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \)

This definition shows us that we can obtain the cotangent by dividing the cosine of an angle by its sine or simply by flipping the value of the tangent. Because the cotangent involves both sine and cosine functions, it carries properties from both.

• It is undefined when \( \sin x = 0 \) because division by zero is undefined. This occurs at integer multiples of \( \pi \), for example, \( 0, \pi, 2\pi \), etc.
• Cotangent is positive in the first and third quadrants of the unit circle and negative in the second and fourth.

Radian Measurement

Radians are a way of measuring angles, like degrees, but more suited for mathematical calculations, especially in trigonometry and calculus. Understanding radians is crucial in evaluating angular functions and expressions such as trigonometric functions.

A radian is based on the radius of a circle. One radian is the angle created when the radius is wrapped along the circle's circumference. This leads to several important properties of radians:

• There are \( 2\pi \) radians in a full circle, as the circumference of a circle is \( 2\pi r \), where \( r \) is the radius.
• To convert degrees to radians, you multiply by \( \frac{\pi}{180} \). Therefore, \( 180^\circ = \pi \) radians.
• Angles bigger than \( \pi \) radians wrap around the circle more than halfway.

When working with trigonometric functions in radians, always ensure your calculator is set to radian mode. This is crucial as using degree mode can lead to incorrect results, especially when evaluating functions like the cotangent.

Reciprocal Identities

Reciprocal identities are a fundamental aspect of trigonometry, helping relate different trigonometric functions through reciprocal relationships.

These identities provide a method to simplify trigonometric expressions and solve equations. The main reciprocal identities are:

• \( \csc x = \frac{1}{\sin x} \) - the reciprocal of sine.
• \( \sec x = \frac{1}{\cos x} \) - the reciprocal of cosine.
• \( \cot x = \frac{1}{\tan x} \) - the reciprocal of tangent.

These identities are particularly handy when one specific trigonometric function is known or when simplifying expressions, as they allow switching between different function forms.

Reciprocal identities are essential in calculus and advanced trigonometry and establish deeper relationships among trigonometric functions. By learning these identities, students can more effectively tackle complex trigonometric evaluations and transformations.