Understanding the Problem: Maximum Value of sin x cos x

We are asked to determine the largest possible value that the trigonometric expression \(\sin x \cdot \cos x\) can attain for any real number \(x\). Finding maximum or minimum values of trigonometric functions often involves using identities to simplify the expression or understanding the range of the functions involved.

Let the expression be \(E = \sin x \cdot \cos x\).

Using Trigonometric Identities for Maximum Value

To find the maximum value of the product \(\sin x \cdot \cos x\), we can use a common trigonometric identity. The double angle identity for sine is particularly useful here:

\(\sin(2x) = 2 \sin x \cos x\)

Notice that the right side of this identity contains the product \(\sin x \cos x\). We can rearrange the identity to isolate this product:

Divide both sides by 2:

\(\frac{1}{2} \sin(2x) = \sin x \cos x\)

So, the expression we are interested in, \(\sin x \cos x\), is equal to \(\frac{1}{2} \sin(2x)\).

Now, finding the maximum value of \(\sin x \cos x\) is equivalent to finding the maximum value of \(\frac{1}{2} \sin(2x)\).

Analyzing the Range of the Sine Function

The sine function, \(\sin \theta\), for any real angle \(\theta\), has a well-defined range of values. The values of \(\sin \theta\) are always between -1 and 1, inclusive. This can be written as:

\(-1 \le \sin \theta \le 1\)

In our expression \(\frac{1}{2} \sin(2x)\), the angle is \(2x\). As \(x\) varies over all real numbers, \(2x\) also varies over all real numbers. Therefore, the value of \(\sin(2x)\) will also be within the standard range of the sine function, i.e., between -1 and 1.

The maximum value that \(\sin(2x)\) can reach is 1.

Calculating the Maximum Value of sin x cos x

We have the relationship \(\sin x \cos x = \frac{1}{2} \sin(2x)\).

To find the maximum value of \(\sin x \cos x\), we take the maximum value of the right side, which depends on the maximum value of \(\sin(2x)\).

Maximum value of \(\sin x \cos x = \frac{1}{2} \times (\text{Maximum value of } \sin(2x))\)

Substitute the maximum value of \(\sin(2x)\), which is 1:

Maximum value of \(\sin x \cos x = \frac{1}{2} \times 1 = \frac{1}{2}\).

Summary

We used the identity \(\sin x \cos x = \frac{1}{2}\sin(2x)\). Since the maximum value of \(\sin(2x)\) is 1, the maximum value of \(\frac{1}{2}\sin(2x)\) is \(\frac{1}{2} \times 1 = \frac{1}{2}\).

Revision Table: Maximum Value Calculation Steps

Additional Information: Trigonometric Functions and Properties

Understanding the properties and graphs of trigonometric functions is essential.

Range of Basic Trigonometric Functions

• Sine (\(\sin \theta\)): The range is \([-1, 1]\). The graph oscillates between -1 and 1.
• Cosine (\(\cos \theta\)): The range is \([-1, 1]\). The graph also oscillates between -1 and 1.
• Tangent (\(\tan \theta\)): The range is \((-\infty, \infty)\). The graph has vertical asymptotes.

Double Angle Identities

The identity \(\sin(2x) = 2 \sin x \cos x\) is part of a set of double angle identities:

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

These identities are derived from the sum formulas (\(\sin(A+B)\), \(\cos(A+B)\), \(\tan(A+B)\)) by setting \(A=B=x\). Using these identities helps in simplifying expressions, solving equations, and finding maximum or minimum values of trigonometric functions.