Problem 50 - Precalculus: A Concise Course

Question

Use the properties of inverse trigonometric functions to evaluate the expression. $$ \tan (\arctan 45) $$

Step-by-Step Solution

Answer

The value of \(\tan (\arctan 45)\) is 45.

Step 1: Identify the Inner Function

First, focus on the function inside the brackets, which is \(\arctan 45\). This is the inverse tangent function of 45.

Step 2: Apply the Inverse Property

Next, apply \(\tan\) to this value. The property of the inverse function allows us to essentially 'cancel out' the \(\tan\) and \(\arctan\). Therefore, \(\tan (\arctan 45) = 45\).

Key Concepts

Evaluating Expressions

Evaluating expressions can seem complex at first, but understanding the steps involved makes it much easier.
Begin by identifying the innermost part of the expression, as it often simplifies the whole process. In our example, the expression is \( \tan (\arctan 45) \), where \( \arctan 45 \) is the inner function.
When faced with such expressions, you need to recognize that inverse trigonometric functions can be simplified using specific properties.
In this situation, it's essential to apply the property that inverse trigonometric functions have with their regular counterparts.
In this way, the evaluation becomes quite straightforward.

Here are some key ideas to remember when evaluating similar expressions:

• Start by breaking down the expression into its core components.
• Identify functions inside the expression, focusing on the inverse trigonometric part when present.
• Apply known properties to simplify the expression as much as possible.

This makes finding the value of such expressions more manageable.

Tangent Function

The tangent function is a fundamental trigonometric function that you often encounter in trigonometry problems.
Typically written as \( \tan(\theta) \), where \( \theta \) is an angle, it is defined using a right triangle as the ratio of the opposite side to the adjacent side.
In the context of the unit circle, the tangent function values are the y-values divided by the x-values of a point on the circle.
The tangent function has a range from negative to positive infinity, but understanding the properties makes it handy in computations.

Some important properties of the tangent function are:

• Periodicity: The tangent function repeats every \( \pi \) radians.
• Vertical asymptotes, where the function is undefined, happen at odd multiples of \( \pi/2 \).
• Sensitivity: Small changes in the angle can cause large changes in the tangent value, especially near the asymptotes.

Knowing these properties helps you manage expressions like \( \tan(\arctan(x)) \), allowing us to see how functions and their inverses interact and simplify neatly.

Trigonometric Identities

Trigonometric identities are essential tools for evaluating and simplifying expressions. They help transform complex trigonometric expressions into simpler forms.
In our example, the identity that most importantly comes into play is the inverse property.

The inverse property states that for a trigonometric function and its inverse, such as \( \tan \) and \( \arctan \), applying the function followed by its inverse, or vice versa, results in the original input.
In simpler terms, it 'cancels out' the effects of the function.
This property allowed us to simplify \( \tan(\arctan 45) \) directly to 45, as the \( \tan \) and \( \arctan \) functions undo each other's effects.

Some other crucial trigonometric identities include:

• Pythagorean identities like \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Angle sum and difference identities, which help with calculating angles that aren't standard like \( \sin(a \pm b) \).
• Double angle formulas such as \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \).

Understanding these identities enables you to simplify a wide range of trigonometric problems effectively.