Problem 36 - Algebra and Trigonometry

Question

Use the Addition Formula for Tangent to prove the Double-Angle Formula for Tangent.

Step-by-Step Solution

Answer

The double-angle formula for tangent is \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\).

Step 1: Recall the Addition Formula for Tangent

The addition formula for tangent is given by: \[ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \] We'll use this formula to derive the double-angle formula for tangent by considering \(a = b = \theta\).

Step 2: Set Up the Double-Angle Identity

Substitute \(a = \theta\) and \(b = \theta\) into the addition formula: \[ \tan(2\theta) = \frac{\tan(\theta) + \tan(\theta)}{1 - \tan(\theta)\tan(\theta)} \] This simplifies to: \[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \] This expression is the double-angle formula for tangent.

Key Concepts

Addition Formula for Tangent

The tangent addition formula is an essential trigonometry identity used to find the tangent of the sum of two angles. It states that for angles \(a\) and \(b\), the tangent is given by: \[ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \]This formula allows us to calculate the tangent of a combined angle by using the tangents of the individual angles. To derive the double-angle formula for tangent, we can apply this logic by letting both angles be the same, i.e., \(a = b = \theta\). This results in:\[ \tan(2\theta) = \frac{\tan(\theta) + \tan(\theta)}{1 - \tan(\theta)\tan(\theta)} \]This is particularly useful as it helps simplify expressions in trigonometry and is widely used in various applications like solving trigonometric equations and proving other identities.

Trigonometry Identities

Trigonometry identities are equations involving trigonometric functions that are true for all values of the included variables. They provide valuable tools for simplifying and solving trigonometric equations. Knowing these identities well can help students manipulate and simplify mathematical expressions involving trigonometric functions.Some of the most common trigonometric identities include:

• Reciprocal identities - like \(\sin(\theta) = \frac{1}{\csc(\theta)}\), which express the relationship between trigonometric functions and their reciprocals.
• Pythagorean identities - such as \(\sin^2(\theta) + \cos^2(\theta) = 1\), which stem from the Pythagorean theorem applied to a unit circle.
• Double-angle identities - these include, for example, the double-angle formula for tangent which we derived using the addition formula.

Understanding these identities is key to mastering concepts in trigonometry, as they act as building blocks for more complex equations and provide insight into the properties of trigonometric functions.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the ratios of its sides. The main functions include sine, cosine, and tangent, which are foundational concepts in trigonometry.

• Sine \((\sin)\) - Represents the ratio of the opposite side to the hypotenuse of a right-angled triangle.
• Cosine \((\cos)\) - Represents the ratio of the adjacent side to the hypotenuse.
• Tangent \((\tan)\) - Represents the ratio of the opposite side to the adjacent side. It's the function that often uses the addition and double-angle formulas.

Trigonometric functions are periodic, meaning they repeat values in regular intervals, which is crucial for solving equations involving periodic phenomena in fields like engineering, physics, and even music. They also extend beyond right triangles to apply in unit-circle-based definitions, allowing for calculation of angles greater than 90 degrees.