Problem 53 - Algebra 2

Question

Use double-angle identities to write each expression, using trigonometric functions of \(\theta\) instead of 4\(\theta .\) $$ \tan 4 \theta $$

Step-by-Step Solution

Answer

The expression for \( \tan(4\theta) \) in terms of \( \tan\theta \) using double-angle identities is: \[ \tan 4\theta = \frac{2 \frac{2 \tan \theta}{1 - \tan^2 \theta}}{1 - \left(\frac{2 \tan \theta}{1 - \tan^2 \theta}\right)^2} \]

Step 1: Understand the double-angle formulas

The double angle identities for tangent are \[ \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \]. This identity allows the expression of \( \tan(2\theta) \) in terms of \( \tan\theta \). However, the prompt asks for \( \tan(4\theta) \). We'll need to apply the double angle identity twice.

Step 2: Apply the double-angle identity first time

Firstly, consider \( 4\theta = 2(2\theta) \). So, \( \tan(4\theta) = \tan[2(2\theta)] \). Apply the double-angle identity to get \[ \tan[2(2\theta)] = \frac{2\tan(2\theta)}{1 - \tan^2(2\theta)} \]. Now we have expressed \( \tan(4\theta) \) in terms of \( \tan(2\theta) \), but we need it in terms of \( \tan\theta \).

Step 3: Apply the double-angle identity second time

In Step 2, we have got \( \tan(4\theta) = \frac{2\tan(2\theta)}{1 - \tan^2(2\theta)} \). Now, apply the double-angle identity to \( \tan(2\theta) \) in both the numerator and denominator. So, we get \[ \tan(4\theta) = \frac{2[\frac{2\tan(\theta)}{1 - \tan^2(\theta)}]}{1 - [\frac{2\tan(\theta)}{1 - \tan^2(\theta)}]^2} \]. After simplifying this, we get the expression for \( \tan(4\theta) \) in terms of \( \tan\theta \), fulfilling the prompt requirements.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are essential components of mathematics, particularly in geometry and calculus.
They express relationships between angles and sides of triangles. The main trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions are crucial for various calculations related to circles, waves, and oscillations.

Understanding these functions is the foundation for solving complex problems involving angles and side lengths, such as those found in the field of trigonometry. It's important to understand how these functions interrelate and how they can be transformed.
For example:

• The sine and cosine functions are used to determine the length of sides and the angle of elevation.
• The tangent function, on the other hand, relates the opposite and adjacent sides of a right triangle.
• Double-angle identities use these trigonometric functions to transform expressions involving multiple angles into simpler terms.

Mastering these trigonometric functions and their identities, such as double-angle identities, allows us to simplify complex expressions.
This helps in expanding our understanding and application of various mathematical concepts.

Understanding the Tangent Function

The tangent function (\( \tan \)) is one of the primary trigonometric functions and plays a significant role in trigonometry.
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
Mathematically, it is expressed as:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]
The tangent function is unique because it can often yield very large values and is undefined at certain angles, specifically every \(90^\circ\) increment.
This occurs because the length of the adjacent side approaches zero, creating a division by zero scenario.

The tangent function is periodic, repeating every \(180^\circ\) (or \(\pid\) radians), and it is helpful when calculating angles in various fields:

• In calculus, tangent helps in determining the slope of curves.
• In physics, it is used to analyze forces and motion.
• In engineering, it helps design and analyze waves and circles.

Understanding and utilizing the tangent function and its transformations is crucial in many mathematical applications, such as when working with double-angle identities.
Let's now explore how transformations of angles affect these functions.

Exploring Angle Transformation

Angle transformation refers to changing an angle measurement in mathematical expressions, an essential part of simplifying trigonometric problems.
When we transform angles, we often use identities such as the double-angle formulas to rewrite expressions in simpler terms.
For instance, in the given problem, the expression for \(\tan(4\theta)\) is rewritten using double-angle identities in terms of \(\tan(\theta)\).

The double-angle identities are particularly useful:

• For tangent, the double angle identity is \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\).
• This identity can be applied repeatedly to transform \(\tan(4\theta)\) into terms of \(\tan(\theta)\).
• This is done by expressing \(\tan(4\theta)\) as \(\tan(2(2\theta))\), then applying the identity twice.

Transforming angles in this manner is not only beneficial for simplifying expressions but also makes calculations more efficient.
By expressing trigonometric functions in terms of simpler angles, we reduce the complexity and potential errors in computation.
Angle transformation using identities like the double-angle formula enriches our toolkit for solving advanced trigonometric problems.