Problem 22 - Precalculus Mathematics for Calculus

Question

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. $$\tan ^{-1}(-0.25713)$$

Step-by-Step Solution

Answer

The approximate value is \(-0.25141\) radians.

Step 1: Understand the Function

The expression \( an^{-1}(x)\), also known as the inverse tangent or arctan, determines the angle \(\theta\) whose tangent is \(x\). The result of \(\tan^{-1}(x)\) is typically expressed in radians.

Step 2: Input the Value into the Calculator

Using a scientific calculator, input the value \(-0.25713\) into the inverse tangent function. Make sure the calculator is in radian mode, as the result of inverse trigonometric functions is conventionally given in radians.

Step 3: Read the Approximate Value

After inputting \(-0.25713\) into \(\tan^{-1}()\), read the calculated output value from the calculator. This value will be the angle in radians, correct to five decimal places.

Step 4: Record the Result

The calculator should return approximately \(-0.25141\) (or close depending on calculator precision) when rounded to five decimal places. This is the radian measure of the angle whose tangent is \(-0.25713\).

Key Concepts

Understanding the Arctangent Function

The arctangent function, often represented as \( \tan^{-1}(x) \) or \( \text{arctan}(x) \), plays an essential role in trigonometry. It's the inverse of the tangent function, specifically designed to find an angle when the tangent value is known. When you see \( \tan^{-1}(x) \), it asks, "What angle has a tangent of \( x \)?"

Here's a quick guide on the arctangent function:

• The output is an angle, typically expressed in radians.
• The angles returned by arctangent will fall within the range of \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
• It's very useful in real-world applications, such as finding angles in right-angled triangles.

This function is crucial when solving problems involving angle measurements and converting between different trigonometric identities. It allows us to move seamlessly from ratios back to angles.

Calculator Use for Inverse Trigonometric Functions

Using a calculator effectively is a major part of solving trigonometric exercises. To find the approximate value of \( \tan^{-1}(-0.25713) \) using a calculator, follow these straightforward steps:

• Ensure your calculator is switched to "radian mode." This step is crucial because the standard output for inverse trigonometric functions is in radians.
• Locate the "tan" function on your calculator, which often shares a button with "arctan" or "tan\(^{-1}\)." You might need to press a "shift" or "2nd" function key first.
• Carefully enter the number \(-0.25713\), and then press the inverse tan button to compute the value.

Rounding: The result will likely show many decimal places. For accuracy, it's common to round the result to five decimal places. In our case, the answer rounds to approximately \(-0.25141\).
Utilizing your calculator correctly ensures precision and speeds up complex trigonometric calculations.

Radian Measure in Trigonometry

Radian measure is a fundamental concept in trigonometry and the wider field of mathematics. Unlike degrees, which divide a circle into 360 parts, radians provide a mathematical approach based on arc length:

• One radian equals the angle made by an arc with the same length as the circle's radius.
• A circle measures \(2\pi\) radians, equivalent to 360 degrees. This means \(\pi\) radians equals 180 degrees.
• This unit simplifies equations and is preferred in most mathematical contexts, particularly calculus.

Understanding and working in radians is crucial, as it ensures you accurately interpret outputs from functions like \( \tan^{-1}(x) \). The number \(-0.25141\), for example, signifies an angle in radian measure, and it holds significance in defining the rotation or the directed angle based on the given trigonometric situation.