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Chapter 6: Problem 43

Simplify each expression. $$\frac{\sin (2 x)}{2}+\sin (2 x)$$

Short Answer

Expert verified
\frac{3}{2} \text{sin}(2x)

Step by step solution

01

Factor out the common term

Identify the common term in the expression \( \frac{\text{sin}(2x)}{2} + \text{sin}(2x) \). The common term is \( \text{sin}(2x) \). Factor \( \text{sin}(2x) \) out of the expression: \[ \text{sin}(2x) \times \frac{1}{2} + \text{sin}(2x) \times 1 \]
02

Simplify the expression inside the parentheses

Combine the terms inside the parentheses: \[ \text{sin}(2x) \times \bigg( \frac{1}{2} + 1 \bigg) \]
03

Add the fractions

Add the fractions \( \frac{1}{2} + 1 \): \[ \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \]
04

Finalize the simplification

Multiply \( \text{sin}(2x) \) by the simplified fraction: \[ \text{sin}(2x) \times \frac{3}{2} = \frac{3}{2} \text{sin}(2x) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Trigonometric Functions
Sometimes in trigonometric expressions, you will need to factor out common terms to simplify.
In the given exercise, the expression is \( \frac{\sin (2 x)}{2}+\sin (2 x) \).
Notice that both terms include \( \sin (2x) \).
This allows us to factor \( \sin (2x) \) out.
You rewrite the expression as:
\( \sin (2x) \times \frac{1}{2} + \sin (2x) \times 1 \).
Factoring is useful because it simplifies expressions and makes complex calculations easier to handle.
Sine Function
The sine function, denoted as \( \sin(x) \), is one of the fundamental trigonometric functions.
It comes from the study of right-angle triangles.
For a given angle in a right triangle, \( \sin(x) \) represents the ratio of the length of the side opposite the angle to the hypotenuse.
In our context, \( \sin(2x) \) means we are taking the sine of twice the angle.
Doubling the angle is a common occurrence in more complex trigonometric problems and usually requires simplification steps as shown.
Fraction Addition
Adding fractions is a crucial step in many math problems, including this one.
In the simplified trigonometric expression, we find:
\( \frac{1}{2} + 1 \).
To add these fractions, first rewrite 1 as a fraction: \( \frac{2}{2} \).
Now the sum becomes:
\( \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \).
Knowing how to find a common denominator is necessary to combine fractions successfully.
Simple steps like this help streamline and simplify more complex mathematical expressions.

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