Question

Simplify. $$\sin 3 \theta \cos 2 \theta-\cos 3 \theta \sin 2 \theta$$

Short answer

\(\sin(\theta)\)

Step-by-step solution

Identify the Trigonometric Identity

The given expression resembles the right-hand side of the sine difference identity, which is \(\sin (A) \cos (B) - \cos (A) \sin (B) = \sin (A - B)\). By comparing it to this identity, we can identify \(A = 3\theta\) and \(B = 2\theta\).

Apply the Sine Difference Identity

Using the identity \(\sin(A - B)\) where \(A = 3\theta\) and \(B = 2\theta\), we substitute these values into the identity to get the simplified form \(\sin(3\theta - 2\theta)\).

Simplify the Inside of the Sine Function

Subtract the values inside the sine function to get \(\sin(\theta)\).

Key concepts

Trigonometric Identities

Understanding trigonometric identities is key to mastering the art of simplifying complex trigonometric expressions. A trigonometric identity is an equation that holds true for all values within the domains of the variables present. These identities are relationships between trigonometric functions that are based on the properties of a right-angled triangle or the unit circle.

One of the fundamental identities is the sine difference identity, which is essential for simplifying expressions like \(\sin 3\theta \cos 2\theta - \cos 3\theta \sin 2\theta\). This identity states that \(\sin(A) \cos(B) - \cos(A) \sin(B) = \sin(A - B)\), where A and B are angles in radians or degrees. These identities are not just useful for academic exercises but are also applicable in various fields such as physics, engineering, and computer science where waveforms and oscillations are analyzed. By internalizing these identities, you set the foundation for solving more advanced trigonometric problems.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a frequent task in mathematics, often requiring the application of trigonometric identities. The objective is to transform a complex expression into a more manageable form, making it easier to interpret or solve. For instance, in the exercise given, we are presented with \(\sin 3\theta \cos 2\theta - \cos 3\theta \sin 2\theta\), which may initially seem daunting.

To simplify, we first recognize patterns that match known identities. In the mentioned expression, the pattern matches the sine difference identity. By substituting the corresponding angles into the identity, we obtain a simpler expression, \(\sin(\theta)\), making further calculations or evaluations much simpler. A clear understanding of how and when to use these identities allows for an efficient and methodical approach to simplifying trigonometric expressions.

Sine Function

The sine function is one of the primary trigonometric functions and plays a significant role in various areas of math and science. Defined initially for right-angled triangles, \(\sin(\theta)\) denotes the ratio of the length of the opposite side to that of the hypotenuse, where \(\theta\) is one of the non-right angles.

In a broader sense, at any angle \(\theta\), sine corresponds to the y-coordinate of a point on the unit circle. This function is periodic and oscillates between -1 and 1. It's essential when describing wave patterns, circular motion, and harmonic oscillations.

Applying the Sine Function to Simplify

Following identification and application of the sine difference identity, we arrive at \(\sin(\theta)\). This expression is much simpler and more meaningful, especially when considering the function's application in real-world problems such as determining the height of an object based on its distance and angle of elevation. Thus, understanding the sine function isn't just about memorizing its shape or basic properties, but also about applying it to simplify complex expressions and solve practical problems.