Problem 29 - Precalculus with Limits

Question

In Exercises 29-36, use a double-angle formula to rewrite the expression. \( 6 \sin x \cos x \)

Step-by-Step Solution

Answer

Using the double-angle formula, \( 6 \sin x \cos x \) can be rewritten as \( 3 \sin(2x) \).

Step 1: Identify the relevant double-angle formula

Examine the available double angle formulas. There is a specific one that relates sin(2x) to the expression \( \sin x \cos x \). The formula is: \( \sin(2x) = 2 \sin x \cos x \).

Step 2: Rearrange the double-angle formula

According to Step 1, we have the double angle formula connecting the terms sin and cos. We can re-arrange this formula to look like the given expression. The rearranged formula looks like this: \( d \sin x \cos x = \sin(2x) \), where d is 2 in our formula, but can be any constant.

Step 3: Apply the double-angle formula to the expression

We apply the above formula to our expression \(6 \sin x \cos x \). With d = 6 in our case, we can divide the expression by 2 to match the constant in the double-angle formula. That gives us \(3 \sin(2x)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations that involve trigonometric functions like sine and cosine. They provide relationships between these functions and can simplify complex trigonometric expressions. These identities are key in solving various types of trigonometry and calculus problems.

Some of the common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal Identities: \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \)
• Quotient Identities: \( \tan x = \frac{\sin x}{\cos x} \)

Understanding these identities helps in recognizing patterns and solving expressions like \( 6 \sin x \cos x \), which can be simplified using the double-angle formula for sine.

Trigonometric identities not only make calculations easier but also deepen comprehension of how trigonometric functions behave and relate to one another.

sin(2x)

The expression \( \sin(2x) \) refers to the sine of a double angle. This is a useful concept in trigonometry for simplifying expressions and solving equations. Understanding how to apply it is crucial for solving various math problems.

According to the double-angle formula for sine, \( \sin(2x) = 2 \sin x \cos x \).

This formula is derived from the addition formulas of sine. It shows that instead of calculating \( \sin \) of twice an angle directly, we can express it in terms of \( \sin \) and \( \cos \) of the original angle \( x \). This is especially useful when dealing with integration or differentiation involving trigonometric functions.

In our exercise, the expression \( 6 \sin x \cos x \) was transformed into \( 3 \sin(2x) \) using this very concept. This type of manipulation is common in trigonometric simplifications and is a neat trick to have in your math toolkit.

Trigonometry

Trigonometry is a branch of mathematics that deals with the study of triangles, particularly right triangles. The main focus is on relationships between the angles and the sides of triangles.

• Sine (\( \sin \))
• Cosine (\( \cos \))
• Tangent (\( \tan \))

These three are the primary trigonometric functions, useful in various branches of science and engineering.

In trigonometry, understanding how angles relate to each other is critical. Double-angle formulas, such as the one for \( \sin(2x) \), help express trigonometric functions of multiple angles more easily. They are applied in simplifying expressions, solving equations, and analyzing wave functions.

The exercise involving \( 6 \sin x \cos x \) demonstrated how a combination of sine and cosine can be compactly represented using a double-angle formula, revealing the interconnected nature of trigonometric operations. This interconnection allows for solving a wide array of mathematical problems efficiently.