Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\tan \left(45^{\circ}+x\right)$$

Step-by-Step Solution

Answer

\( \tan(45^{\circ} + x) = \frac{1 + \tan x}{1 - \tan x} \).

Step 1: Recall the Tangent Addition Formula

The tangent addition formula states that \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). In this case, we have \( a = 45^{\circ} \) and \( b = x \).

Step 2: Find the Tangent of 45 Degrees

We know that \( \tan 45^{\circ} = 1 \) because the tangent of 45 degrees is equal to 1.

Step 3: Apply the Tangent Addition Formula

Substitute \( \tan 45^{\circ} = 1 \) and \( \tan x = \tan x \) into the tangent addition formula:\[\tan(45^{\circ} + x) = \frac{1 + \tan x}{1 - 1 \cdot \tan x}\]This simplifies to\[\tan(45^{\circ} + x) = \frac{1 + \tan x}{1 - \tan x}\]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships in the world of mathematics that relate the angles and sides of triangles. These identities are especially useful in simplifying expressions and solving equations involving trigonometric functions such as sine, cosine, and tangent.
The tangent function, denoted as \(\tan(\theta)\), is one of the primary trigonometric functions. This function represents the ratio of the opposite side to the adjacent side in a right-angled triangle. Trigonometric identities, like the tangent addition formula, help in finding the values of trigonometric functions for compound angles.
Understanding and memorizing trigonometric identities allows students to transform complex expressions into simpler forms. This is particularly helpful in solving problems where the desired function needs to be expressed using a single variable or simplified significantly.

• Sine, Cosine, and Tangent are core trigonometric functions.
• Identities relate these functions to each other and can simplify complex expressions.
• The Tangent Addition Formula is a specific identity used for altering compound angles.

Angle Sum Identities

Angle Sum Identities are formulas that allow us to find the trigonometric function values for sums of two angles. These are invaluable when dealing with expressions involving angles such as \(a + b\).
The most common Angle Sum Identities are for sine, cosine, and tangent functions, enabling the calculation of these functions when given the sum or difference of two angles. The tangent addition formula, for instance, is expressed as \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \).
When you know individual angle values or their trigonometric functions, these identities allow you to find the compound value without memorizing new facts.

• Apply these identities to simplify expressions like \(\tan(45^{\circ}+x)\).
• Essential for overcoming problems involving angle operations.
• Enable conversion into a single argument expression.

Mathematical Functions

Mathematical functions are like machines that take an input, perform a specific operation, and produce an output. In trigonometry, functions like \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\) process angles to give ratios related to right triangles.
Understanding mathematical functions in trigonometry involves recognizing how these functions perform calculations based on certain rules, such as those provided by trigonometric identities. For example, \(\tan(x)\) is processed using both its definition and identities like the tangent addition formula to find its value for certain inputs.
In the original problem, the use of the tangent addition formula converts the input \(45^{\circ} + x\) into a functional form that can be more easily evaluated or graphically represented.

• Functions take inputs and mathematically transform them.
• Trigonometric functions provide specific angle ratio outputs.
• Conversions lead to simpler solution pathways.