Problem 12 - Algebra and Trigonometry

Question

Use a calculator to find an approximate value of each expression rounded to five decimal places, if it is defined. $$ \tan ^{-1}(-4) $$

Step-by-Step Solution

Answer

The approximate value of \( \tan^{-1}(-4) \) is -1.32582 when rounded to five decimal places.

Step 1: Understand the Inverse Tangent Function

The inverse tangent function, denoted as \( \tan^{-1}(x) \), is used to determine the angle whose tangent is \( x \). The range of \( \tan^{-1}(x) \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Step 2: Input into Calculator

Enter the value of -4 into a calculator with an inverse tangent or arctan function. Use the \( \tan^{-1} \) or \( \arctan \) button on the calculator.

Step 3: Calculate and Round

The calculator will provide the value of the angle in radians. Round this value to five decimal places to find an approximate result.

Step 4: Interpret the Calculator Output

The calculator outputs \( \tan^{-1}(-4) \approx -1.32582 \) radians, which means the angle that has a tangent of -4 is approximately -1.32582 when rounded to five decimal places.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are essential in studying periodic phenomena and have applications in fields like physics, engineering, and signal processing.
Among the core trigonometric functions are sine, cosine, and tangent. These are often abbreviated as sin, cos, and tan, respectively. Tangent, in particular, is defined as the ratio of the opposite side to the adjacent side of a right-angle triangle. It is expressed as:

• \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)

This is key to understanding how the inverse tangent function works. Knowing how each function operates and adds value, studying angles and how they interact with various triangle facets is fundamental for effectively tackling problems like the original exercise.

Inverse Functions

Inverse functions essentially reverse what the original function does. For trigonometric functions, the inverse seeks to find the angle that results in a particular trigonometric ratio.
The inverse tangent function, represented as \( \tan^{-1} \) or arctan, specifically gives the angle whose tangent is a given number. For example, if we know \( \tan(\theta) = x \), then \( \theta = \tan^{-1}(x) \).
This concept is crucial because it allows us to transition from a known ratio back to the angle measurement, especially valuable in fields that require precise angle calculations. Understanding the inverse function's role explains why \( \tan^{-1}(-4) \) asks us to find the angle with a tangent of \(-4\). This angle will be within the range \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), which helps pinpoint the exact or approximate angle for real-life applications.

Calculator Usage

Using a calculator for trigonometric functions can simplify many problems. Most scientific calculators have a function to find inverse trigonometric values, which is accessible via specific keys labeled as \( \tan^{-1} \) or \( \arctan \).
To solve a problem like \( \tan^{-1}(-4) \), follow these steps:

• Turn on the calculator.
• Locate and press the inverse tangent button \( \tan^{-1} \) or \( \arctan \).
• Enter the number \(-4\).
• Press the equals or enter button to compute the result.

The calculator provides the angle in radians, which can be quite precise. In educational or applied settings, rounding the result to five decimal places enhances clarity and usefulness. For \( \tan^{-1}(-4) \), this returns \(-1.32582\) radians. Understanding how to leverage these functions on a calculator is beneficial for students needing to verify their manual calculations or solve complex problems quickly.