Problem 25 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \tan \left(\tan ^{-1} 1.2\right) $$

Step-by-Step Solution

Answer

The exact value is 1.2.

Step 1: Understanding the Problem

We need to find the exact value of the expression \( \tan (\tan^{-1} 1.2) \). This expression involves an inverse trigonometric function and its complementary trigonometric function.

Step 1: Analyzing the Expression

The expression \( \tan (\tan^{-1} 1.2) \) means we find the tangent of the angle whose tangent is 1.2. The inverse tangent function, \( \tan^{-1} \), gives us an angle whose tangent is the specified value inside the function.

Step 2: Applying Trigonometric Properties

By definition, if \( y = \tan^{-1} x \), then \( \tan(y) = x \). For the given expression \( \tan(\tan^{-1} 1.2) \), \( x = 1.2 \). Thus, this simplifies the expression as \( \tan(y) = 1.2 \).

Step 3: Conclusion

Since the tangent function \( \tan \) and its inverse \( \tan^{-1} \) are complementary, applying the function \( \tan \) to its inverse simply returns the original value inside the inverse function, which is 1.2. So, \( \tan (\tan^{-1} 1.2) = 1.2 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angle whose trigonometric function value is given. They are the tools to "undo" the effect of a trigonometric function. For the tangent function, the inverse is known as the arctangent, written as \( \tan^{-1} \). When you use \( \tan^{-1}(x) \), you're essentially asking: "Which angle has a tangent value of \( x \)?" This function is particularly useful because it allows us to translate a trigonometric ratio back into an angle measure.

• Arctangent Function \( \tan^{-1} \): For a given value, it provides the angle \( y \) such that \( \tan(y) = x \).
• Range of Arctangent: The function \( \tan^{-1}(x) \) typically returns angles in the range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
• Real-World Application: Used in situations where angles need to be determined from gradient or slope measurements.

To summarize, inverse functions play a critical role in trigonometry as they help bridge the gap between trigonometric values and angle measures.

Tangent Function

The tangent function \( \tan \) relates an angle in a right triangle to the ratio of the length of the opposite side to the adjacent side. Unlike sine and cosine, the tangent function can take on all real values because its range is \((-\infty, +\infty)\).

• Behavior: The tangent function repeats every \( \pi \) radians, meaning it has a period of \( \pi \).
• Asymptotes: Tangent has vertical asymptotes where cosine equals zero, at odd multiples of \( \frac{\pi}{2} \).
• Graphical Representation: It's helpful to visualize \( \tan(x) \) as a series of repeating curves moving vertically between \(-\infty\) and \(+\infty\).

This function is critical in both analytical and geometrical contexts, given its ability to handle all real numbers and repeatedly exhibit a straightforward pattern.

Exact Value in Trigonometry

Finding the exact value of a trigonometric function means determining the precise trigonometric ratio without approximations. In contexts where inverse trigonometric functions are involved, such as \( \tan(\tan^{-1}(x)) \), simplification often reveals that the value simplifies back to the known ratio. In this exercise:

• \( \tan(\tan^{-1}(1.2)) \)

This expression simplifies directly to 1.2, because \( \tan \) and \( \tan^{-1} \) are inverse functions.
Here, the concept of exact values highlights how trigonometric and inverse functions cancel each other out, returning the original value inside the function. It's similar to how a square root and a square are inverse operations in algebra.