Problem 27 - Precalculus with Calculus Previews

Question

Use a double-angle formula to write the given expression as a single trigonometric function of twice the angle. $$ \frac{\tan 3 t}{1-\tan ^{2} 3 t} $$

Step-by-Step Solution

Answer

\( \tan 6t \).

Step 1: Recognize the Formula

The expression \( \frac{\tan x}{1 - \tan^2 x} \) is known as the tangent double-angle identity. The formula for \( \tan 2x \) is given by \( \frac{2 \tan x}{1 - \tan^2 x} \).

Step 2: Identify the Angle

In the problem, \( x = 3t \). Therefore, you are working with \( \tan 2(3t) \), which simplifies to \( \tan 6t \).

Step 3: Apply the Double-Angle Formula

Since \( \frac{\tan 3t}{1 - \tan^2 3t} \) matches the form \( \frac{2 \tan x}{1 - \tan^2 x} \) for the angle \( x = 3t \), this expression simplifies to \( \tan 6t \).

Step 4: Simplify the Expression

Therefore, the given expression \( \frac{\tan 3t}{1 - \tan^2 3t} \) simplifies using the double-angle formula to the single trigonometric function \( \tan 6t \).

Key Concepts

Trigonometric Identities

Trigonometric identities are like handy tools that help us manipulate and simplify trigonometric expressions. They often involve relationships between different trigonometric functions—like sine, cosine, and tangent—and are super useful when solving equations or proving other mathematical statements. One crucial set of identities is the double-angle identities. These allow us to express trigonometric functions of double angles. For tangent, the double-angle identity tells us that:

• \( \tan 2x = \frac{2\tan x}{1 - \tan^2 x} \)

These identities can condense more complex expressions into simpler forms, which is exactly what they do in the given exercise. Recognizing a trigonometric expression as part of a known identity is often the first step in simplifying or solving it.

Tangent Function

The tangent function is one of the primary trigonometric functions and is often symbolized as \( \tan(x) \). It takes an angle as its input and gives a ratio of the opposite side to the adjacent side in a right triangle. Unlike sine and cosine, which have values strictly between -1 and 1, the tangent function can have any real value.For the tangent double-angle identity,

• Understanding \( \tan x \) helps simplify the expression \( \frac{\tan x}{1 - \tan^2 x} \).
• When you plug in the angle \( x = 3t \) into the tangent double-angle formula, you're finding the tangent of twice that angle.

Thus, from the given formula, \( \tan 6t \) is found. This reflects how powerful understanding the tangent function and its properties can be in simplifying problems.

Angle Manipulation

Manipulating angles can be seen as a fundamental aspect of working with trigonometric functions, allowing us to express and solve equations in terms of different angle representations. In trigonometric identities, manipulating an angle by doubling it lets us take complex equations and compress them into more manageable forms.With the double-angle formula, we take the angle \( 3t \) and use it as part of our identity. Here’s how it works:

• You start with \( x = 3t \), then consider \( 2x = 6t \).
• Using the identity, you actually change the problem from an expression in terms of \( 3t \) to one in \( 6t \). This helps simplify problems when you know the value of \( t \) or when working within known trigonometric boundaries or conditions.

By converting between these angles, we derive simpler results without altering the integrity or outcome of the problem.