Expressing Trigonometric Functions As A Single Angle Tangent Addition And Subtraction Formulas

In the realm of trigonometry, expressing trigonometric functions as a single angle is a fundamental skill that unlocks a multitude of problem-solving possibilities. This often involves employing trigonometric identities, particularly the tangent addition and subtraction formulas. These formulas provide a powerful mechanism to consolidate expressions involving tangents of multiple angles into a single, more manageable term. In this comprehensive guide, we will delve into the intricacies of these formulas, explore their applications, and demonstrate how they can be used to simplify complex trigonometric expressions.

Understanding the Tangent Addition and Subtraction Formulas

The tangent addition and subtraction formulas are cornerstones of trigonometric identities, allowing us to express the tangent of the sum or difference of two angles in terms of the individual tangents of those angles. These formulas are mathematically expressed as:

$$\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$

Where:

• A and B represent any two angles.
• The "$\pm$" symbol indicates that we use the top sign for addition and the bottom sign for subtraction.

These formulas elegantly capture the relationship between the tangent of a combined angle and the tangents of its constituent angles. Understanding and applying these formulas is crucial for simplifying trigonometric expressions and solving a wide range of trigonometric problems.

Derivation of the Tangent Addition Formula

To appreciate the elegance of the tangent addition formula, let's briefly outline its derivation. We begin with the sine and cosine addition formulas:

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$

$$\cos(A + B) = \cos A \cos B - \sin A \sin B$$

Now, recall that $\tan x = \frac{\sin x}{\cos x}$. Therefore:

$$\tan(A + B) = \frac{\sin(A + B)}{\cos(A + B)} = \frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}$$

To express this in terms of tangents, we divide both the numerator and denominator by $\cos A \cos B$:

$$\tan(A + B) = \frac{\frac{\sin A \cos B}{\cos A \cos B} + \frac{\cos A \sin B}{\cos A \cos B}}{\frac{\cos A \cos B}{\cos A \cos B} - \frac{\sin A \sin B}{\cos A \cos B}}$$

Simplifying, we get:

$$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

Derivation of the Tangent Subtraction Formula

The derivation of the tangent subtraction formula follows a similar approach. We start with the sine and cosine subtraction formulas:

$$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

$$\cos(A - B) = \cos A \cos B + \sin A \sin B$$

Then,

$$\tan(A - B) = \frac{\sin(A - B)}{\cos(A - B)} = \frac{\sin A \cos B - \cos A \sin B}{\cos A \cos B + \sin A \sin B}$$

Dividing both the numerator and denominator by $\cos A \cos B$:

$$\tan(A - B) = \frac{\frac{\sin A \cos B}{\cos A \cos B} - \frac{\cos A \sin B}{\cos A \cos B}}{\frac{\cos A \cos B}{\cos A \cos B} + \frac{\sin A \sin B}{\cos A \cos B}}$$

Simplifying, we arrive at:

$$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

Applying the Tangent Addition and Subtraction Formulas: A Step-by-Step Approach

Now that we have a firm grasp of the formulas themselves, let's explore how to apply them effectively. The general strategy involves recognizing expressions that match the form of the formulas and then substituting the appropriate values.

Step 1: Identify the Pattern

The first step is to carefully examine the given expression and see if it resembles either the tangent addition or subtraction formula. Look for the following key features:

• A fraction with a numerator containing the sum or difference of two tangent terms ($\tan A \pm \tan B$).
• A denominator containing 1 plus or minus the product of the same two tangent terms ($1 \mp \tan A \tan B$).

If you can identify these patterns, you're well on your way to applying the formulas.

Step 2: Determine the Values of A and B

Once you've identified the pattern, the next step is to determine the values of the angles A and B that correspond to the terms in the expression. This usually involves carefully matching the terms in the expression to the terms in the formula.

Step 3: Substitute into the Formula

With the values of A and B in hand, simply substitute them into the appropriate formula (either addition or subtraction) and simplify. This will express the original expression as the tangent of a single angle.

Illustrative Example: Simplifying Trigonometric Expressions

Let's consider the specific example provided:

$$\frac{\tan 55^{\circ} + \tan 17^{\circ}}{1 - \tan 55^{\circ} \tan 17^{\circ}}$$

Step 1: Identify the Pattern

We can clearly see that this expression matches the form of the tangent addition formula:

$$\frac{\tan A + \tan B}{1 - \tan A \tan B}$$

Step 2: Determine the Values of A and B

By comparing the expression to the formula, we can identify:

• A = 55°
• B = 17°

Step 3: Substitute into the Formula

Substituting these values into the tangent addition formula, we get:

$$\tan(55^{\circ} + 17^{\circ}) = \frac{\tan 55^{\circ} + \tan 17^{\circ}}{1 - \tan 55^{\circ} \tan 17^{\circ}}$$

Simplifying the angle, we obtain:

$$\tan(72^{\circ}) = \frac{\tan 55^{\circ} + \tan 17^{\circ}}{1 - \tan 55^{\circ} \tan 17^{\circ}}$$

Therefore, the expression $\frac{\tan 55^{\circ} + \tan 17^{\circ}}{1 - \tan 55^{\circ} \tan 17^{\circ}}$ can be expressed as $\tan 72^{\circ}$.

Practice Problems: Sharpening Your Skills

To solidify your understanding, let's tackle a few more practice problems.

Problem 1

Express the following as a function of a single angle:

$$\frac{\tan 80^{\circ} - \tan 20^{\circ}}{1 + \tan 80^{\circ} \tan 20^{\circ}}$$

Solution:

This expression matches the tangent subtraction formula. We have A = 80° and B = 20°. Substituting into the formula, we get:

$$\tan(80^{\circ} - 20^{\circ}) = \frac{\tan 80^{\circ} - \tan 20^{\circ}}{1 + \tan 80^{\circ} \tan 20^{\circ}}$$

Simplifying, we find:

$$\tan(60^{\circ}) = \frac{\tan 80^{\circ} - \tan 20^{\circ}}{1 + \tan 80^{\circ} \tan 20^{\circ}}$$

Problem 2

Express the following as a function of a single angle:

$$\frac{\tan 40^{\circ} + \tan 5^{\circ}}{1 - \tan 40^{\circ} \tan 5^{\circ}}$$

Solution:

This expression aligns with the tangent addition formula. We have A = 40° and B = 5°. Substituting into the formula:

$$\tan(40^{\circ} + 5^{\circ}) = \frac{\tan 40^{\circ} + \tan 5^{\circ}}{1 - \tan 40^{\circ} \tan 5^{\circ}}$$

Simplifying:

$$\tan(45^{\circ}) = \frac{\tan 40^{\circ} + \tan 5^{\circ}}{1 - \tan 40^{\circ} \tan 5^{\circ}}$$

Advanced Applications and Problem-Solving Techniques

The tangent addition and subtraction formulas extend beyond simple simplification. They are invaluable tools in solving trigonometric equations, proving identities, and tackling more complex problems involving angles and their relationships. Here are some advanced applications and techniques:

Solving Trigonometric Equations

These formulas can be used to solve equations where the unknown angle is embedded within a tangent function of a sum or difference. By applying the formulas, we can often break down the equation into a more manageable form, allowing us to isolate the unknown.

Proving Trigonometric Identities

The tangent addition and subtraction formulas are frequently used as building blocks in proving more complex trigonometric identities. By strategically applying these formulas and other known identities, we can demonstrate the equivalence of different trigonometric expressions.

Angle Sum and Difference Identities in Calculus

In calculus, these identities play a crucial role in differentiating and integrating trigonometric functions. They enable us to express complex trigonometric integrals and derivatives in simpler terms, making calculations more tractable.

Conclusion: Mastering Trigonometric Transformations

The tangent addition and subtraction formulas are powerful tools in the trigonometric toolbox. By understanding their derivations and mastering their applications, you can significantly enhance your ability to simplify trigonometric expressions, solve equations, and prove identities. Whether you're a student delving into trigonometry for the first time or a seasoned mathematician tackling complex problems, these formulas will undoubtedly prove their worth. Embrace their power, practice their application, and unlock the full potential of trigonometric transformations.

By mastering these formulas, you gain a deeper understanding of the interconnectedness of trigonometric functions and their relationships. This knowledge empowers you to approach trigonometric problems with confidence and elegance. So, continue exploring, continue practicing, and continue expanding your trigonometric horizons.