Problem 37 - Calculus Single Variable

Question

Use one or more of the basic trigonometric identities to derive the given identity. \(\tan (\theta+\phi)=\frac{\tan (\theta)+\tan (\phi)}{1-\tan (\theta) \tan (\phi)}\)

Step-by-Step Solution

Answer

To derive \( \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \), use the sine and cosine sum identities, express tangent as sine over cosine, simplify, and factor.

Step 1: Understand the identity

The identity to be derived is the tangent sum formula: \( \tan(\theta + \phi) = \frac{\tan(\theta) + \tan(\phi)}{1 - \tan(\theta) \tan(\phi)} \). This formula expresses the tangent of a sum of two angles in terms of the tangents of the individual angles.

Step 2: Recall related trigonometric identities

Recall the sine and cosine sum identities: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \) and \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). These identities are essential in expressing \( \tan(a + b) \) in terms of \( \sin \) and \( \cos \).

Step 3: Express tangent using sine and cosine

The tangent of an angle can be expressed as the ratio of sine to cosine: \( \tan a = \frac{\sin a}{\cos a} \). Substituting into the tangent of a sum, we write \( \tan(\theta + \phi) = \frac{\sin(\theta + \phi)}{\cos(\theta + \phi)} \).

Step 4: Substitute sine and cosine identities

Using the sine and cosine sum identities, substitute to find \( \tan(\theta + \phi) = \frac{\sin \theta \cos \phi + \cos \theta \sin \phi}{\cos \theta \cos \phi - \sin \theta \sin \phi} \).

Step 5: Rewrite using tangent

Rewrite the sine and cosine expressions in terms of tangent: \( \sin \theta = \tan \theta \cos \theta \) and \( \sin \phi = \tan \phi \cos \phi \). Substitute these into the equation.

Step 6: Simplify the expression

Substitute \( \sin \theta = \tan \theta \cos \theta \) and \( \sin \phi = \tan \phi \cos \phi \) into the equation: \( \tan(\theta + \phi) = \frac{\tan \theta \cos \theta \cos \phi + \cos \theta \tan \phi \cos \phi}{\cos \theta \cos \phi - \tan \theta \cos \theta \tan \phi \cos \phi} \).

Step 7: Factorize and cancel

Factor \( \cos \theta \cos \phi \) from numerator and denominator: \( \tan(\theta + \phi) = \frac{\cos \theta \cos \phi (\tan \theta + \tan \phi)}{\cos \theta \cos \phi (1 - \tan \theta \tan \phi)} \). This results in \( \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \) after canceling \( \cos \theta \cos \phi \).

Key Concepts

Tangent Sum Formula

Understanding the tangent sum formula can be quite enlightening when dealing with trigonometry. The tangent sum formula states: \( \tan(\theta + \phi) = \frac{\tan(\theta) + \tan(\phi)}{1 - \tan(\theta) \tan(\phi)} \)This equation is pivotal to simplify expressions where two angles are added together and you want to find their tangent. By utilizing this formula, you can break down complex trigonometric problems into simpler pieces.

• The numerator \( \tan(\theta) + \tan(\phi) \) indicates the sum of the tangents of the two angles individually.
• The denominator \( 1 - \tan(\theta) \tan(\phi) \) adjusts the sum for any interaction between the two angles.

Using this approach, we can see how separate angles come together in trigonometry using their tangent. It serves as a balance by combining both additive and multiplicative factors of these tangents.

Sine and Cosine Identities

The sine and cosine identities are some of the most important tools in trigonometry. They allow you to decompose and simplify expressions involving trigonometric functions. Let's look at two key identities:

• For any angles \(a\) and \(b\), the sine sum identity is: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• Similarly, the cosine sum identity is: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

These identities help in expressing complicated trigonometric functions into more manageable forms by breaking down the functions into sine and cosine terms.
These are then used as foundational steps for many derivations, such as our tangent sum formula. They enable us to substitute and express tangent in terms of sine and cosine, hence fostering greater understanding of relationships between different angles.

Derivation of Trigonometric Identities

Deriving trigonometric identities might seem a bit daunting at first, but it actually follows a logical step-by-step process. The derivation involves expressing complex trigonometric functions using known identities. In the example of deriving the tangent sum formula:

• The strategy is to express tangent as a ratio of sine and cosine: \( \tan a = \frac{\sin a}{\cos a} \)
• Use the sine and cosine sum identities to express \(\tan(\theta + \phi)\): \( \tan(\theta + \phi) = \frac{\sin(\theta + \phi)}{\cos(\theta + \phi)} \)
• Substitute the identities for sine and cosine sums into the equation.
• Simplify by expressing \(\sin \theta\) and \(\sin \phi\) in terms of \(\tan \theta\) and \(\tan \phi\).
• Finally, factor and cancel terms to reveal the sum formula.

This process not only helps in understanding how the formulas are built but also in enhancing a deeper understanding of the relationships between trigonometric functions. You get to see how all identities are interlinked, revealing the underlying structure of trigonometry.