Problem 58 - Precalculus : Building Concepts and Connections

Question

In Exercises \(39-60,\) use a scientific calculator to evaluate the given trigonometric functions to four decimal places. $$ \cot \left(-217^{\circ}\right) $$

Step-by-Step Solution

Answer

The value of \(\cot (-217^{\circ})\) is approximately -0.5773.

Step 1: Converting Negative Angle

First, we need to convert the negative angle -217° to its corresponding positive angle. As angles in circular functions are periodic with period 360°, we can add 360°. Therefore, the positive equivalent angle for -217° is -217°+360° = 143°.

Step 2: Calculating Cotangent

Since cotangent is the reciprocal function of tangent, to evaluate \(\cot (143^{\circ})\), we will calculate \(\tan (143^{\circ})\) first using a calculator, and then take the reciprocal.

Step 3: Evaluating Cotangent

The evaluation using a scientific calculator gives the result as approximately -0.5773. Please note that solution can slightly vary depending on the precision of calculator.

Key Concepts

Understanding Cotangent

Cotangent is one of the six fundamental trigonometric functions crucial for understanding relationships in right triangles and circular functions. Specifically, cotangent is defined as the reciprocal of tangent for a given angle. This means that if we know the tangent of an angle, the cotangent is simply one divided by the tangent. For example, if \[\tan(\theta) = x\]then:\[\cot(\theta) = \frac{1}{x}\]
This relationship becomes particularly useful in many geometric and trigonometric calculations. The cotangent function can help determine ratios of side lengths in right triangles, and it also plays roles in various calculus, physics, and engineering applications.

• In a right triangle, cotangent relates the adjacent side to the opposite side for a given angle.
• It is periodic with a period of 180 degrees or \( \pi \) radians, meaning its values repeat every 180 degrees.

Understanding the cotangent helps in solving trigonometric equations and understanding the mirrored relationship of angles across critical thresholds like 90 degrees and 180 degrees.

Reciprocal Functions and Their Role

Reciprocal functions are an integral part of trigonometry and algebra. The basic idea of a reciprocal function is simple: it is formed by taking the reciprocal (or the inverse) of another function's output. This means if you have a function \( f(x) \), the reciprocal function is \( \frac{1}{f(x)} \). In the context of trigonometric functions, these relationships are foundational.
Cotangent, cosecant, and secant are the reciprocal functions in trigonometry, opposing tangent, sine, and cosine respectively:

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

These reciprocal relationships allow for a different perspective and approach to solving problems that involve trigonometric functions. Reciprocal functions provide alternative solutions and insights that can simplify complex expressions or equations in mathematical analysis.

Angle Conversion Simplified

Angle conversion is essential when working with trigonometric functions because angles can be expressed in different ways. Typically, angles are measured in degrees or radians. However, sometimes within trigonometric calculations, you might need to convert between negative and positive angles to simplify expressions or align with calculator settings. Conversion is quite straightforward due to the periodic nature of trigonometric functions.

• Every full rotation signifies 360 degrees (or \(2\pi\) radians), which is why adding or subtracting 360 degrees often results in equivalent angles.
• To convert a negative angle to a positive angle, add 360 degrees until the result is positive.
• For radians, add \(2\pi\) until achieving a positive value.
  

In practical terms, angle conversion ensures that analysis in both circular motion and wave functions accurately reflects changes without altering the fundamental properties. Though it might seem a simple idea, being comfortable with angle conversion is essential, especially when solving problems or using a calculator set in degree mode.