Problem 25 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\tan \left(\tan ^{-1} 5\right)\)

Step-by-Step Solution

Answer

The value is 5.

Step 1: Understanding the Inverse Function

The expression \( \tan (\tan^{-1} 5) \) involves the tangent and its inverse function. The inverse function \( \tan^{-1} \) is defined such that \( \tan^{-1}(x) \) gives the angle whose tangent is \( x \). Therefore, applying \( \tan \) to \( \tan^{-1}(x) \) returns the original value \( x \), as long as \( x \) is within the range of \( \tan^{-1} \).

Step 2: Applying the Property of Inverse Trigonometric Functions

Using the property \( \tan(\tan^{-1}(x)) = x \) for all real numbers \( x \), we directly evaluate \( \tan (\tan^{-1} 5) \). Since 5 is a real number, this property applies, and the result is \( 5 \).

Key Concepts

Understanding the Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the basic trigonometric functions. It relates the angles of a right triangle to the ratio of two of its sides. Specifically, in a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. This can be written in the formula:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
It's important to remember that the tangent function is periodic with a period of \( \pi \), meaning it repeats its values every \( \pi \) units. Additionally, unlike functions like sine and cosine, the tangent function can output any real value, which is significant in understanding its range and domain properties.

• In trigonometry, \( \tan \) is often used to find angles or side lengths in various problems.
• It is essential to grasp tangent's behavior to solve equations involving angles and interpret its graphical representation.

Exploring the Tangent Inverse Function

The tangent inverse function, written as \( \tan^{-1} x \) or \( \arctan x \), serves as the inverse of the tangent function. This means it takes a real number as input and delivers an angle whose tangent value returns the original number.
The range of \( \tan^{-1} x \) is usually from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). In simpler terms, it provides angles located in the first and fourth quadrants of the unit circle.

• This function is critical when determining the angle for a given tangent value.
• Its function relies on many properties and identities that simplify trigonometric calculations.
• Using \( \tan \) with \( \tan^{-1} \) as in \( \tan(\tan^{-1}(x)) \) returns \( x \) when \( x \) is within the correct range, simplifying complex expressions.

The Role of Trigonometric Identities

Trigonometric identities are equations that are valid for all values relevant to their expression. They are useful in simplifying trigonometric expressions and solving equations. They form the backbone of most trigonometric calculations and transformations.
Some common trigonometric identities include:

• Pythagorean Identities: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Sum and Difference Formulas: Used to calculate the sine, cosine, and tangent of sums or differences of angles.
• Double Angle Formulas: Useful for breaking down expressions into handleable parts, such as \( \sin 2\theta = 2 \sin \theta \cos \theta\).

These identities are essential in calculus, physics, engineering, and other fields that require precise trigonometric calculations.

Working with Real Numbers in Trigonometry

Real numbers encompass all the numbers on the number line, including both rational and irrational numbers. They are the most common way to represent quantities and have immense significance in trigonometry. Most trigonometric functions, including the tangent, can take and output real numbers, making them versatile in numerous applications.
Using real numbers in trigonometry allows for resolving expressions like \( \tan(\tan^{-1}(x)) \) as shown above. Such expressions are defined for all real numbers due to the vast range of the tangent function. This property allows seamless conversion from angles to linear measures and vice versa, a frequent necessity in problem-solving.
Understanding real numbers is crucial when dealing with trigonometric functions because:

• They ensure that trigonometric values remain consistent and valid across various calculations.
• They allow the modeling of physical phenomena using trigonometric equations.

In essence, grasping real-number properties helps navigate through trigonometry with ease and accuracy.