Problem 49 - Algebra 2

Question

Rewrite each expression as a trigonometric function of a single angle measure. $$ \frac{\tan 3 \theta-\tan \theta}{1+\tan 3 \theta \tan \theta} $$

Step-by-Step Solution

Answer

\(\tan (2\theta)\

Step 1: Identify the identity

We should identify the used identity. The given expression is in the form of \(\frac{\tan a - \tan b}{1 + \tan a \tan b}\), where \(a = 3\theta\) and \(b = \theta\) which is the formula for \(\tan (a - b)\). Hence the given expression represents \(\tan (a - b)\).

Step 2: Apply the identity

Apply the identity \(\tan (a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\) to the given expression. Substituting \(a = 3\theta\) and \(b = \theta\) into the identity we get \(\tan (3\theta - \theta)\).

Step 3: Simplify the expression

The expression \(\tan (3\theta - \theta)\) simplifies to \(\tan (2\theta)\). This is the single-angle trigonometric function that the original expression simplifies to.

Key Concepts

Angle Simplification

When working with trigonometric expressions, simplifying them to a function of a single angle can make calculations easier and more straightforward. Here, we are given a trigonometric expression involving angles 3\(\theta\) and \(\theta\). By using established trigonometric identities, we can simplify this expression to an equivalent one involving a single angle.

• The expression \(\frac{\tan 3\theta-\tan \theta}{1+\tan 3\theta \tan \theta}\) resembles a known identity, specifically the tangent subtraction formula.
• By recognizing this structure, we can simplify it to one single-angle trigonometric function, namely \(\tan 2\theta\).

This process not only consolidates the expression but also makes further mathematical operations simpler.

Tangent Difference Formula

The tangent difference formula is a key trigonometric identity used to simplify expressions involving the difference of angles. It allows us to transform the difference of two tangent expressions into a single tangent value.

• The formula is given by: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
• In the context of this exercise, \(a\) is substituted by \(3\theta\) and \(b\) by \(\theta\), leading to the expression: \[ \tan(3\theta - \theta) = \frac{\tan 3\theta - \tan \theta}{1 + \tan 3\theta \tan \theta} \]

The identity simplifies the expression by reducing two terms (\(3\theta\) and \(\theta\)) into a single term, effectively compressing the complexity involved.

Trigonometric Functions

Understanding trigonometric functions like tangent, sine, and cosine is essential in mathematics, especially when dealing with angles and periodic phenomena. Each of these functions has unique properties and identities that allow for simplified expressions and calculations.

• For tangent, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), making it a useful tool for expressing angles as ratios.
• Trigonometric identities, such as the tangent difference formula, capitalize on these properties to transform expressions into more manageable forms.

In this exercise, by using the tangent's specific properties and its difference formula, we reduced a complicated expression into \(\tan 2\theta\), demonstrating the power and efficiency of trigonometric identities in simplifying complex problems.