Problem 19 - Algebra and Trigonometry

Question

\(11-22\) . Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. \(\tan ^{-1}(1.23456)\)

Step-by-Step Solution

Answer

The result is approximately 0.88844 radians.

Step 1: Understanding the Problem

We need to find the value of the inverse tangent (or arctangent) of 1.23456. The notation \( \tan^{-1}(x) \) means 'the angle whose tangent is \( x \)'. Use a calculator to compute this value.

Step 2: Using a Calculator

Enter 1.23456 into your calculator, then use the inverse tangent function, usually labeled as \( \tan^{-1} \) or "atan". Ensure that your calculator is set to the correct angle unit (degrees or radians) as required for your exercise.

Step 3: Obtaining the Result

After entering \( \tan^{-1}(1.23456) \) into the calculator, note the result. The value is approximately 0.88844 radians when rounded to five decimal places.

Key Concepts

Arctangent

The arctangent, also represented as \( \tan^{-1}(x) \) or "atan," is an inverse trigonometric function. It produces the angle whose tangent is the specified number \( x \). This function is fundamental when we want to determine an angle given a tangent value. It maps a real number to an angle, usually within the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians.
Understanding when and how to use arctangent is essential in mathematical problem-solving. It provides a way to reverse the tangent operation, which is especially useful in geometry, trigonometry, and calculus. For example, if you know the opposite and adjacent side lengths of a right triangle, you can use the tangent and then arctangent to find the angle between the two sides.
When solving problems, it's crucial to understand that the output of arctangent is an angle and to interpret the result accordingly based on the unit in use (degrees or radians).

Tangent

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. Mathematically, this is represented as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
Tangent is one of the primary trigonometric functions and is widely used in various fields such as physics, engineering, and architecture. It helps in understanding angles and their relationships with side lengths in triangles.
While working with tangent, ensure you remember:

• Tangent values can be positive or negative depending on the quadrant of the angle.
• It is undefined for angles where the cosine is zero (e.g., 90 and 270 degrees in standard position).

Recognizing these properties allows for better comprehension in problem-solving scenarios involving trigonometric functions.

Calculator Use

Calculators are invaluable tools for quickly solving problems involving complex functions like trigonometric equations. They come equipped with functions that can easily compute things like square roots, powers, and trigonometric functions including sine, cosine, and tangent (as well as their inverses).
When finding the arctangent of a number, you should:

• Turn on your calculator and ensure it is in the correct mode (degrees or radians).
• Input the number you need to get the inverse tangent for, in this case, \(1.23456\).
• Use the \(\tan^{-1}\) or "atan" button to calculate the angular result.

Remember that different calculators may have slight variations in operation or appearance but will generally have similar functionality when performing these operations.

Radians

Radians are a unit of angular measurement based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians are based on the ratio of the circumference of the circle to its radius. The whole circle is \(2\pi\) radians, meaning \(360\) degrees is equivalent to \(2\pi\) radians. Therefore, \(180\) degrees equal \(\pi\) radians.
When working with radians, it is often more naturally linked to calculus due to its direct relationship with the geometry of a circle. This makes it particularly useful in mathematics and physics.
Key points you should keep in mind about radians:

• Always check if your calculator is set to degrees or radians, especially when working with trigonometric functions, as seen in this problem with \(\tan^{-1}(1.23456)\).
• Converting between degrees and radians can be done using: degrees \(= \frac{180}{\pi} \times \text{radians}\) and vice versa.

Understanding radians enhances your capability to perform more complex calculations and allows for a stronger grasp of mathematical principles.