https://file.aiquickdraw.com/mathbot/file/1726580152326gf5xqoyi.jpg

We are given the following expression, assuming \(x\) and \(y\) are small and measured in radians:

\[
\sin^2(x) - \sin(6y) + \frac{\sin(2x) - \sin(y)}{4}
\]

We will now use small-angle approximations for trigonometric functions, which are valid when \(x\) and \(y\) are small. The small-angle approximations are:

\[
\sin(x) \approx x \quad \text{and} \quad \cos(x) \approx 1
\]

### Step-by-step evaluation:

1. **First term: \(\sin^2(x)\)**  
   Since \(\sin(x) \approx x\) for small \(x\), we have:
   \[
   \sin^2(x) \approx x^2
   \]

2. **Second term: \(\sin(6y)\)**  
   Using the small-angle approximation \(\sin(6y) \approx 6y\), we get:
   \[
   \sin(6y) \approx 6y
   \]

3. **Third term: \(\frac{\sin(2x) - \sin(y)}{4}\)**  
   Using the approximations \(\sin(2x) \approx 2x\) and \(\sin(y) \approx y\), we have:
   \[
   \frac{\sin(2x) - \sin(y)}{4} \approx \frac{2x - y}{4}
   \]

### Putting it all together:

\[
\sin^2(x) - \sin(6y) + \frac{\sin(2x) - \sin(y)}{4} \approx x^2 - 6y + \frac{2x - y}{4}
\]

Now simplifying the third term:

\[
x^2 - 6y + \frac{2x - y}{4} = x^2 - 6y + \frac{2x}{4} - \frac{y}{4} = x^2 - 6y + \frac{x}{2} - \frac{y}{4}
\]

Thus, the expression is approximately:

\[
x^2 + \frac{x}{2} - 6y - \frac{y}{4}
\]

This is the simplified approximation.

Would you like further details on any specific part, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. How does the small-angle approximation work for other trigonometric functions like cosine?
2. What is the importance of the small-angle approximation in physics and engineering?
3. How would the result differ if higher-order approximations were used for the sine function?
4. Could you apply a similar approximation for \(\tan(x)\) in a different expression?
5. How would you evaluate this expression if \(x\) and \(y\) were not small?

**Tip**: The small-angle approximation is particularly useful in many real-world problems involving oscillations and waves where the angles involved are small enough for the approximation to hold.

Given that x and y are small and measured in radians, what is sin^2(x) - sin(6y) + (sin(2x) - sin(y)) / 4 approximately equal to?

Explore the use of small-angle approximations to simplify trigonometric expressions. This guide explains how to approximate sin^2(x), sin(6y), and (sin(2x) - sin(y)) / 4 using small values of x and y, suitable for college-level mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation