Problem 64 - Precalculus : Building Concepts and Connections

Question

Evaluate the given expressions to four decimal places with a calculator. $$\cot ^{-1}(-1.8)$$

Step-by-Step Solution

Answer

The value of \(\cot^{-1}(-1.8)\) to four decimal places is -0.4877 radians or -27.9454 degrees, depending on the mode (radian or degree) your calculator is set to.

Step 1: Understand the problem

We are asked to evaluate the inverse cotangent of -1.8. This means we are looking for the angle whose cotangent is -1.8. Remember that cotangent is the reciprocal of the tangent.

Step 2: Use a calculator

Use a scientific calculator to compute the inverse cotangent of -1.8. Most calculators might not have a direct function for \(cot^{-1}\), but remember that \(cot^{-1}(x) = tan^{-1}(1/x)\), so you can calculate the inverse tangent of the reciprocal of -1.8. To do this, simply divide 1 by -1.8 and compute the inverse tangent of the resulting decimal.

Step 3: Round the result

Round the result from the calculator to four decimal places. This should give the final result.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when we know the trigonometric ratios. These functions are the inverse of the basic trigonometric functions, such as sine, cosine, and tangent, but their names include 'arc', like \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\). Another notation used is 'arcsin', 'arccos', and 'arctan'. These functions help us solve for an angle given a known ratio, which is essentially what inverse cotangent does. In the original problem, you're asked to evaluate \(\cot^{-1}(-1.8)\), which translates to finding the angle whose cotangent is -1.8. It's important to note that the range of the inverse trigonometric functions is limited to deliver principal values. For instance, \(\tan^{-1}\) returns angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which assists in providing a single unique solution for each ratio.

Cotangent Function

The cotangent function is one of the six fundamental trigonometric functions, often less mentioned but equally essential. Just like the tangent function, the cotangent function relates an angle in a right triangle to the ratio of the adjacent side length to the opposite side length.In its simplest explanation, the cotangent is the reciprocal of the tangent function. Mathematically, it can be expressed as:- \(\cot(\theta) = \frac{1}{\tan(\theta)}\)- It can also be written as \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)However, when dealing with the inverse cotangent, or \(\cot^{-1}(x)\), we aim to find an angle such that the cotangent of this angle gives us x. This requires transforming the problem into the more commonly addressed inverse tangent function because many calculators don't provide a direct way to compute \(\cot^{-1}\). Instead, we use the relationship \(\cot^{-1}(x) = \tan^{-1}(1/x)\) to indirectly calculate it using available functions.

Scientific Calculator Usage

Using a scientific calculator to solve for inverse trigonometric functions can simplify many problems, including finding \(\cot^{-1}\). Many scientific calculators are equipped with functions for \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\), but they might not have \(\cot^{-1}\) directly available. To compute \(\cot^{-1}(-1.8)\), follow these steps:

• First, find the reciprocal of -1.8, which is \(-\frac{1}{1.8}\).
• Next, use your calculator to find \(\tan^{-1}\) of this value. \(\text{tan}^{-1}\) is a function typically available on most scientific calculators.
• Finally, the resulting angle should be rounded to four decimal places to meet the precision requirement of your problem.

Formulas for inverse trigonometric functions may need to be input with parentheses on a calculator, so be careful not to make syntax errors. This step-wise approach ensures that your calculations are accurate and comply with expected mathematical practices.