Problem 44 - Algebra 2

Question

Rewrite each expression as a trigonometric function of a single angle measure. $$ \sin 2 \theta \cos \theta+\cos 2 \theta \sin \theta $$

Step-by-Step Solution

Answer

The simplified expression for the given trigonometric function is \( 3 \sin\theta \cos^2\theta - \sin^3\theta \).

Step 1: Apply Double-Angle Identities

The given expression is \( \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta \). The double-angle identities are \( \sin 2\theta = 2 \sin\theta \cos\theta \) and \( \cos 2\theta = \cos^2\theta - \sin^2\theta \) or \( \cos 2\theta = 2\cos^2\theta - 1 \) or \( \cos 2\theta = 1 - 2\sin^2\theta \). For simplicity, the substitution \( \cos 2\theta = \cos^2\theta - \sin^2\theta \) will be used.

Step 2: Substitute the Identities into the Expression

Substitute \( \sin 2\theta = 2 \sin\theta \cos\theta \) and \( \cos 2\theta = \cos^2\theta - \sin^2\theta \) into the expression, which gives \( 2 \sin\theta \cos^2\theta + (\cos^2\theta - \sin^2\theta) \sin\theta \).

Step 3: Simplify the expression

Simplify the expression to get \( 2 \sin\theta \cos^2\theta + \sin\theta \cos^2\theta - \sin^3\theta \) = \( 3 \sin\theta \cos^2\theta - \sin^3\theta \).

Key Concepts

Double-Angle Identities

Double-angle identities are an essential part of trigonometry. They help us relate trigonometric functions when doubling an angle. These identities can simplify expressions and solve equations with different forms of trigonometric functions.
One of the key double-angle identities for sine is \( \sin 2\theta = 2 \sin \theta \cos \theta \). This identity expresses the sine of double the angle \( \theta \) in terms of \( \sin \theta \) and \( \cos \theta \).
For cosine, there are multiple forms you can use, such as:

• \( \cos 2\theta = \cos^2\theta - \sin^2\theta \)
• \( \cos 2\theta = 2\cos^2\theta - 1 \)
• \( \cos 2\theta = 1 - 2\sin^2\theta \)

These forms provide flexibility depending on the given expression and what you're trying to achieve. By using these identities, complex trigonometric expressions can be rewritten to become more manageable and powerful tools for solving problems.

Trigonometric Simplification

Trigonometric simplification is the process of reducing complex trigonometric expressions to simpler forms. This technique is particularly useful in solving equations and proving identities. For the expression \( \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta \), using substitution and simplification helps in reducing the expression.
After applying the double-angle identities, you substitute them into the expression. This changes it into \( 2 \sin \theta \cos^2 \theta + (\cos^2 \theta - \sin^2 \theta) \sin \theta \). In this step, combining like terms and simplifying the expression is crucial.
Arrive at a formula like \( 3 \sin \theta \cos^2 \theta - \sin^3 \theta \), which is easier to use in further calculations or analysis. Simplifying involves:

• Substituting known identities
• Combining like terms
• Using algebraic manipulations for reduction

These steps transform the expression into a single trigonometric function, making problem-solving much easier.

Expressions in Trigonometry

Trigonometric expressions are combinations of trigonometric functions and angles. They can range from simple to highly complex forms and often require manipulation to solve or simplify.
Common trigonometric functions include \( \sin \theta \), \( \cos \theta \), and \( \tan \theta \), among others. In exercises such as \( \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta \), expressing the function in terms of a single angle is an essential task. This helps in understanding and solving equations efficiently.
When dealing with expressions:

• Identify known identities or formulas that can be applied
• Use standard mathematical operations to simplify or transform
• Focus on turning the expression into a single function of one angle

Grasping the mechanics of working with trigonometric expressions opens doors to advanced calculus and analytical geometry topics. Each problem becomes a logic puzzle that requires both trigonometric understanding and algebraic skill to solve. By mastering expressions, students can tackle a wide array of mathematical challenges effectively.