Problem 28 - Precalculus Mathematics for Calculus

Question

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 expression simplifies to \( \frac{3}{2} \).

Step 1: Understand the Expression

The expression given is \( \tan \left(\tan^{-1}\left(\frac{3}{2}\right)\right) \). This involves the arctangent (inverse tangent) and the tangent functions.

Step 2: Simplify the Inner Expression

The inner expression is \( \tan^{-1}\left(\frac{3}{2}\right) \), which represents an angle whose tangent value is \( \frac{3}{2} \).

Step 3: Apply the Tangent Function

Since \( \tan(\theta) = \frac{3}{2} \) for the angle \( \theta = \tan^{-1}\left(\frac{3}{2}\right) \), applying the tangent to this arc tangent gives us the original value back. Thus, \( \tan \left( \tan^{-1}\left(\frac{3}{2}\right) \right) = \frac{3}{2} \).

Step 4: Conclusion

The given composition of the tangent and inverse tangent functions simplifies directly to \( \frac{3}{2} \) because the \( \tan^{-1} \) function specifically returns an angle whose tangent is the provided value.

Key Concepts

Tangent Function

The tangent function, commonly denoted as \( \tan(\theta) \), is a fundamental concept in trigonometry. It relates the ratio of two sides of a right triangle: the side opposite the angle \( \theta \) to the side adjacent to \( \theta \). Visually, if you imagine a right triangle, the tangent of an angle tells you how steeply the opposite side inclines with respect to the adjacent side. When expressed in terms of the angle, the formula is:

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

The tangent function is periodic and repetitive, featuring a period of \( \pi \) radians or \( 180^\circ \). This means that \( \tan(\theta + n\pi) = \tan(\theta) \) for any integer \( n \).
Because of this periodicity, the tangent function extends beyond right triangles and applies to any real-number angle by using the unit circle.
In the unit circle, the tangent is represented by the line that touches the circle at one point and extends vertically. Its positive and negative values repeat as the circle is traversed.

Trigonometry

Trigonometry is the study focused on the relationships between the sides and angles of triangles. This field of mathematics is used extensively in various applications, from simple geometry to high-level engineering and physics. The building blocks of trigonometry are its three primary functions: sine, cosine, and tangent, all derived from right triangles and the unit circle.

• Sine and Cosine Functions: Represent the ratios of the sides in a right triangle and are defined on the unit circle.
• Tangent Function: Relates the opposite side to the adjacent side, and it's particularly useful for determining slopes.

Trigonometry can also delve into more complex concepts, such as:

• Inverse Trigonometric Functions: These functions, like \( \tan^{-1} \), allow you to find angles when you know the sides.
• Trigonometric Identities: Equations like \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) help simplify and solve problems.

Understanding trigonometry provides the tools not just for solving triangles but also for analyzing waves, modeling periodic phenomena, and much more, making it an essential area of study.

Function Composition

Function composition involves applying one function to the results of another. It’s a critical concept when dealing with expressions such as \( \tan(\tan^{-1}(x)) \), where the output of one function becomes the input of another.The notation \( (f \circ g)(x) \) reads as "\( f \) composed with \( g \) of \( x \)" and means \( f(g(x)) \). This allows us to nest functions and evaluate them more sequentially.
In the provided exercise, composition helps simplify the expression \( \tan(\tan^{-1}(\frac{3}{2})) \). The inverse function \( \tan^{-1} \left(\frac{3}{2}\right) \) returns an angle, which, when plugged back into the tangent function, returns the original value \( \frac{3}{2} \).
This works because the inverse function, \( \tan^{-1} \), specifically finds an angle whose tangent is \( \frac{3}{2} \). Applying the original function, \( \tan \), here 'undoes' the inverse, reaffirming the original ratio.