Understanding and Calculating the Period of sin x cos x

When dealing with trigonometric functions, the period is a fundamental concept that describes the smallest positive value P such that the function repeats its values every P units. For the function sin x cos x, we will explore how to derive its period through trigonometric identities and properties.

Deriving the Period of sin x cos x

First, let's explore a trigonometric identity to simplify the expression sin x cos x into a more manageable form. Using the product-to-sum identities, we have:

sin x cos x (1/2) [sin(x x) - sin(x - x)] (1/2) [sin(2x) - sin(0)]

Since sin(0) 0, this simplifies to:

sin x cos x (1/2) sin(2x)

Next, using the identity sin(A) sqrt(2) sin(A π/4), we can further simplify the expression:

sin x cos x (1/2) sqrt(2) sin(2x π/4) (1/√2) sin(2x π/4)

Here, the term 2x π/4 indicates a phase shift, but does not alter the period of the function. The period of the sine function sin(Bx) is given by 2π/|B|. In our case, B 2, so:

Period 2π/2 π

Therefore, the period of sin x cos x is π/2.

Confirming the Period through Function Values

To verify this conclusion, let's examine the values of sin x cos x for specific angles:

When x 0, sin 0 cos 0 1

When x π/2 (90°), sin π/2 cos π/2 1

When x π (180°), sin π cos π -1

When x 3π/2 (270°), sin 3π/2 cos 3π/2 -1

And when x 2π (360°), sin 2π cos 2π 1

The values repeat every π/2, confirming that the period is indeed π/2.

Graphical Representation

A graphical representation using an online graphing tool like Desmos clearly shows the periodic nature of the function:

Figure 1: Graph of sin x cos x showing the periodicity.

The graph demonstrates the function repeats every π/2.

Summary

The period of the function sin x cos x is π/2. This can be derived using trigonometric identities and properties, as well as by observing the periodic behavior of the function.

Key points:

Trigonometric Identity: sin x cos x (1/2) [sin(2x)] Period Calculation: 2π/2 π Periodicity Observation: Values repeat every π/2

Therefore, understanding and calculating the period of trigonometric functions like sin x cos x is crucial for both theoretical and practical applications in mathematics and physics.