Problem 28 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\tan \left(\tan ^{-1}\left(\frac{3}{2}\right)\right)\)

Step-by-Step Solution

Answer

The value is \( \frac{3}{2} \).

Step 1: Understand the Function

The expression involves the inverse tangent function (or arctan) and the tangent function. Specifically, we are looking at the composition of those two functions: \[ \tan \left( \tan^{-1} \left( \frac{3}{2} \right) \right) \] Understanding the properties of these functions, particularly how they cancel each other out, is key.

Step 2: Recall the Inverse Function Property

The inverse tangent function \( \tan^{-1}(x) \) gives an angle whose tangent is \( x \). Therefore, applying the tangent function to its inverse yields the original value within its principal range. That is, for any \( x \) within the range of \( \tan^{-1} \), \[ \tan(\tan^{-1}(x)) = x \] for \( x \) in \((-\infty, \infty)\).

Step 3: Apply the Property to the Expression

Given the expression \( \tan \left( \tan^{-1} \left( \frac{3}{2} \right) \right) \), apply the property of inverse functions from Step 2:\[ \tan \left( \tan^{-1} \left( \frac{3}{2} \right) \right) = \frac{3}{2} \] This is because \( \frac{3}{2} \) is a value for which the inverse tangent function is defined and its result lies in the valid range of the tangent function too.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are incredibly useful in mathematics because they allow us to find angles when given a trigonometric ratio. In other words, if we're given a specific trigonometric value, an inverse function can tell us the angle that would produce that value.
One of the most common inverse trigonometric functions is the inverse tangent function, also known as \( \tan^{-1}(x) \) or arctan. When you see \( \tan^{-1}(x) \), it asks for the angle whose tangent is \( x \).

• For example, if given \[ \tan^{-1}(\frac{3}{2}) \], the function returns the angle whose tangent is \( \frac{3}{2} \).
• These functions return angles typically within specific ranges. For \( \tan^{-1} \), the range is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Understanding the range of inverse functions is crucial because it influences the results you can expect when working with composite functions, like the one in our exercise.

Tangent Function

The tangent function, often written as \( \tan(x) \), is uniquely matched with the inverse tangent. It takes an angle as input and provides a ratio of the lengths of the opposite side to the adjacent side in a right-angled triangle.

• The tangent of an angle \( \theta \) is expressed as \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
• This function is periodic, meaning it repeats its values at regular intervals, specifically every \( \pi \).
• Importantly, the tangent function's range covers all real numbers, \( (-\infty, \infty) \).

When combined with its inverse, the tangent function plays a key role in reverting a notation back to its original format. So, \( \tan(\tan^{-1}(x)) = x \), as long as \( x \) is within the range that \( \tan^{-1} \) returns. This concept directly relates to solving expressions like the one given in the exercise.

Function Properties

Function properties of trigonometric functions, particularly inverses, help us solve mathematical problems efficiently. One of the primary properties is how functions and their inverses interact with each other.

• For instance, if you apply the tangent function after the inverse tangent, you will return to your original input if that input was valid for the inverse operation.
• This is due to the cancellation property: \( \tan(\tan^{-1}(x)) = x \) when \( x \) lies within the allowable range for the tangent function.
• Furthermore, because the output of \( \tan^{-1}(x) \) is always an angle, and angles that the tangent function can naturally handle, you can trust that this cancellation will reliably hold true.

With the original exercise, comprehending these properties allows us to immediately identify that \( \tan(\tan^{-1}(\frac{3}{2})) \) simplifies directly to \( \frac{3}{2} \) through this property smoothening complex calculations into simple results.