Problem 11 - Algebra and Trigonometry

Question

Use a calculator to find an approximate value of each expression rounded to five decimal places, if it is defined. $$ \tan ^{-1} 3 $$

Step-by-Step Solution

Answer

The approximate value of \( \tan^{-1} 3 \) is 1.24905.

Step 1: Understand the Problem

We need to find the value of the inverse tangent function, \( \tan^{-1} 3 \), which is the angle whose tangent is 3. This angle is usually denoted in radians.

Step 2: Use a Calculator

Using a scientific calculator, locate the \( \tan^{-1} \) or \( \arctan \) function. Enter the value 3 into the calculator to calculate \( \tan^{-1} 3 \).

Step 3: Approximate the Result

Calculate \( \tan^{-1} 3 \) using the calculator. The approximate value will be displayed in radians. Note the value and make sure it is rounded to five decimal places.

Step 4: Round the Result

The calculator provides the result approximately as 1.24904577 radians. Round this result to five decimal places. The approximate value is 1.24905.

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry. It arises from the ratios of the sides of a right triangle. When you take a right triangle and focus on one of its acute angles, the tangent of that angle (\( \tan \theta \)) is the ratio of the length of the opposite side to the adjacent side.

This function is cyclical and repeats every \( \pi \) radians or 180 degrees. It's interesting to note that the tangent function can take any real number as an input and spits out outputs ranging from -\( \infty \) to \( \infty \).

Inverse trigonometric functions, like \( \tan^{-1} \), are sometimes necessary to find the angle when the tangent value is known. The output of the inverse tangent function, or \( \arctan \), gives the angle in radians that has the specified tangent value. This is what we call solving for an angle in trigonometry – finding \( \theta \) when you know the value of \( \tan \theta \).

Radians

Radians are a unit of angular measurement used frequently in mathematics. They provide a natural way of describing angles based on the radius of a circle. One complete revolution around a circle – 360 degrees – is equal to \( 2\pi \) radians.

To understand why radians are used, imagine wrapping the radius of a circle along its circumference. Doing so for a complete circle gives \( 2\pi \) such lengths, making it convenient for mathematical calculations.

When using inverse trigonometric functions like \( \tan^{-1} \), the result is typically expressed in radians. This is because radians provide an easy framework for working in calculus and other forms of mathematical analysis. Most standard scientific calculators give trigonometric functions results in radians by default.

Scientific Calculator

Scientific calculators are essential tools for solving mathematical problems involving complex computations. They are programmed to handle a variety of mathematical functions, including trigonometric, exponential, and inverse operations.

Using a scientific calculator to find \( \tan^{-1} 3 \) involves locating the correct function key, often labeled as either "tan⁻¹" or "arctan".

• Start by ensuring the calculator is set to use radians if required by the problem.
• Enter the number 3.
• Press the \( \tan^{-1} \) or "arctan" button to compute the angle.
• Read the result from the calculator display.

This process yields the angle in radians that has a tangent of 3, approximated to a certain number of decimal places, which in this case, is five decimal places: 1.24905.