Problem 37 - Precalculus Essentials

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$\tan \left(\tan ^{-1} 125\right)$$

Step-by-Step Solution

Answer

The answer to \(\tan(\tan^{-1}125)\) is 125.

Step 1: Identify the nature of the problem

The given problem is a trigonometric function problem. Specifically, it involves the tangent function and its inverse.

Step 2: Apply the property of inverse functions

One of the properties of inverse functions is that the function followed by its inverse will result in the original input. In other words, if we take any real number, say x, then \(\tan(\tan^{-1}x) = x\). In the given problem, x is 125.

Step 3: Substitute x with the given value

The given problem asks us to find \(\tan(\tan^{-1}125)\). Since the function \(tan^{-1}125\) is included in the function \(\tan\), based on the property in Step 2, we can replace this by 125.

Key Concepts

Inverse Tangent

Understanding the inverse tangent, or \( \tan^{-1}(x) \), is crucial when working with trigonometric functions. The inverse tangent function is widely used to determine the angle whose tangent is a given value. Unlike the regular tangent function that maps an angle to a ratio, the inverse tangent goes the other way around—it maps a ratio back to an angle.
To clarify:

• The function \( \tan(x) \) tells us what the tangent of angle \x\ is.
• In contrast, \ \tan^{-1}(x) \ gives us the angle which has a tangent equal to \x\.

The range of \ \tan^{-1}(x) \ is from \(-\frac{\pi}{2} \) to \(\frac{\pi}{2}\), meaning the angles found are in this interval. This constrained range ensures every input results in a single, unique output, which is a necessity for a function to have an inverse in mathematics.
This inverse aspect makes it extremely handy when resolving angles in various applications, from geometry to calculus.

Tangent Function

The tangent function, denoted as \ \tan(x) \, plays a pivotal role in trigonometry, helping to relate angles to ratios in a right triangle.
When considering a right triangle:

• \( \tan(x) \) is defined as the ratio of the opposite side to the adjacent side.
• In terms of the unit circle, it relates the y-coordinate (opposite side) to the x-coordinate (adjacent side).

These interpretations make the tangent function crucial for solving various trigonometric problems. Unlike sine and cosine, the tangent function is \(\frac{\sin(x)}{\cos(x)}\), giving it unique properties, including undefined values where cosine is zero. Because of this, the tangent function has a periodicity of \(\pi\).
Understanding the behavior and calculation of the tangent function is essential for using it effectively, especially in conjunction with its inverse. This leads us back to the property used in the exercise that states: \( \tan(\tan^{-1}(x)) = x \). Knowing this property can be a huge time-saver in mathematical problems, especially when calculating exact values for complex expressions.

Inverse Functions

Inverse functions reverse the effect of the original function, returning the input from the output. In the context of trigonometry, we encounter several inverse functions like the inverse tangent, sine, and cosine.
In simple terms:

• If a function takes an input \(a\) and maps it to an output \(b\), then the inverse function takes \(b\) and maps it back to \(a\).
• This reciprocation is the essence of inverse functions.

For the problem given, the inverse function property is used where \( \tan(\tan^{-1}(x)) = x \). This property highlights how the tangent and inverse tangent functions interact. Essentially, applying a function followed by its inverse gets you back to where you started. This makes inverse functions powerful tools for solving equations and simplifying expressions.
Recognizing and understanding this property in inverse functions assists in verification of solution accuracy and efficiency in problem-solving techniques, which is why it was pivotal in the original exercise solution.