Problem 35 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The Double-Angle Formula for sine is \( \sin(2\theta) = 2 \sin \theta \cos \theta \).

Step 1: Understand the Addition Formula for Sine

The Addition Formula for sine is given by: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). This formula helps in determining the sine of the sum of two angles.

Step 2: Set Up the Double Angle Expression

To find the Double-Angle Formula for sine, we need to consider the sine of twice an angle, expressed as \( \sin(2\theta) \). Here, we can consider this as \( \sin(\theta + \theta) \).

Step 3: Apply the Addition Formula for Sine

Substitute \( a \) and \( b \) with \( \theta \) in the Addition Formula for sine: \( \sin(\theta + \theta) = \sin \theta \cos \theta + \cos \theta \sin \theta \).

Step 4: Simplify the Expression

The expression \( \sin \theta \cos \theta + \cos \theta \sin \theta \) simplifies to \( 2 \sin \theta \cos \theta \) by combining like terms since \( \sin \theta \cos \theta \) appears twice in the sum.

Step 5: Conclude with the Double-Angle Formula for Sine

The Double-Angle Formula for sine is thus obtained: \( \sin(2\theta) = 2 \sin \theta \cos \theta \). This shows how applying the Addition Formula leads to the Double-Angle Formula.

Key Concepts

Addition Formula for Sine

The Addition Formula for sine is a fundamental trigonometric identity that allows us to find the sine of the sum of two angles, say \(a\) and \(b\). The formula is expressed as: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]This formula is incredibly useful because it breaks down a complex angle into components that are often easier to handle. Knowing this identity allows you to manipulate and simplify trigonometric expressions effectively.
Here’s how it works:

• The sine of a sum, \( \sin(a + b) \), is decomposed into two products: \( \sin a \cos b \) and \( \cos a \sin b \).
• These products represent the interaction between the angles \(a\) and \(b\) through their sine and cosine values.

By using this formula, you can solve many problems involving angle addition, and it's a stepping stone to understanding more complex trigonometric formulas such as the double-angle and half-angle formulas.

Sine Function

The sine function is one of the most basic trigonometric functions, often abbreviated as \( \sin \). It relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse.
The function has a few key properties:

• It is periodic with a period of \(2\pi\), meaning that \( \sin(\theta) = \sin(\theta + 2\pi) \) for any angle \(\theta\).
• It ranges between -1 and 1 for all real numbers.
• It is an odd function, therefore \( \sin(-\theta) = -\sin(\theta) \).

The sine function is widely used in mathematics, physics, engineering, and many applied sciences. Its graph, known as the sine wave, is smooth and continuous, providing an oscillating curve that is fundamental to signal processing and wave theory.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of variables where both sides of the equal sign are defined. They form the basis of solving trigonometric equations and proving relationships between different trigonometric functions.
Some essential identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Addition Formulas: As mentioned, these include the formulas for \( \sin(a + b) \), \( \cos(a + b) \), and more.
• Double-Angle and Half-Angle Formulas: For example, the Double-Angle Formula for sine is \( \sin(2\theta) = 2 \sin \theta \cos \theta \).
• Even-Odd Identities: These tell us that \( \sin(-\theta) = -\sin(\theta) \) and \( \cos(-\theta) = \cos(\theta) \).

By understanding and mastering these identities, you gain powerful tools to solve a wide array of mathematical problems, enabling you to transform complex expressions into simpler, more manageable forms.