Question

Simplify. $$2 \sin 2 \theta \cos 2 \theta$$

Short answer

\(\sin(4\theta)\)

Step-by-step solution

Recognize the Trigonometric Identity

The given expression can be simplified using a double-angle identity for sine, which is \(\sin(2x) = 2\sin(x)\cos(x)\). In the given expression \(2\sin(2\theta)\cos(2\theta)\), the variables match this form if we let \(x = 2\theta\).

Apply the Double-Angle Identity

Replace \(2\sin(2\theta)\cos(2\theta)\) with \(\sin(4\theta)\), which is obtained by doubling the angle in the identity: \(\sin(2(2\theta)) = 2\sin(2\theta)\cos(2\theta)\).

Write the Simplified Result

The expression simplifies to \(\sin(4\theta)\), which is the most simplified form of the given trigonometric expression.

Key concepts

Double-Angle Identity

Understanding the double-angle identity is essential for simplifying many trigonometric expressions. This powerful tool enables us to express trigonometric functions of double angles — that is, angles of the form \(2x\) — in terms of single angles, simplifying calculations and solving problems.

Specifically, the sine double-angle identity is expressed as \(\sin(2x) = 2\sin(x)\cos(x)\). It arises from the sum formula for sine and is a shortcut for expanding expressions without the need to apply the sum formula repeatedly. The double-angle identities also exist for cosine and tangent, respectively:

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

By using these identities, complex trigonometric expressions involving double angles can be transformed into simpler forms, making them more manageable to work with.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a critical skill in mathematics that involves reducing expressions to their simplest form, using trigonometric identities and algebraic manipulation.

Here's an outline of the general process of simplification:

• Identify Trigonometric Identities: Recognize parts of the expression that match known trigonometric identities, such as double-angle, power-reducing, or sum-to-product identities.
• Apply the Identity: Substitute the identified parts with their simpler trigonometric identity equivalents.
• Algebraic Manipulation: After applying the identities, further simplify the expression using algebraic techniques like factoring, combining like terms, and cancelling common factors.

Tools like the Pythagorean identity, \(\sin^2(x) + \cos^2(x) = 1\), and angle sum and difference formulas also frequently come into play in these simplifications. The goal is to express the trigonometric function in the simplest form possible, making it easier to evaluate, graph, or integrate.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry and are essential in describing the relationships between the angles and sides of triangles, particularly right-angled triangles, and in modeling periodic phenomena.

The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), each corresponding to a specific ratio of sides in a right-angled triangle:

• Sine: The ratio of the opposite side to the hypotenuse, \(\sin(\theta) = \frac{opposite}{hypotenuse}\).
• Cosine: The ratio of the adjacent side to the hypotenuse, \(\cos(\theta) = \frac{adjacent}{hypotenuse}\).
• Tangent: The ratio of the opposite side to the adjacent side, \(\tan(\theta) = \frac{opposite}{adjacent}\).

Additional functions, called the reciprocal trigonometric functions, include cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)), which are the reciprocals of the sine, cosine, and tangent functions, respectively.

These functions underpin many concepts in geometry, physics, engineering, and other fields, and are also the base for more complex trigonometric identities and theorems.