Understanding the Sum Formula for Sine

Oct 13, 2025 by wbrum.org 42 views

Introduction

In trigonometry, the sum and difference formulas are essential for simplifying and manipulating trigonometric expressions. One of the most common formulas is the sum formula for sine, which states that $\sin(x+y) = \sin x \cos y + \cos x \sin y$ . In this article, we will explore how to find the correct expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ using the given options.

Understanding the Sum Formula for Sine

The sum formula for sine is a fundamental concept in trigonometry that allows us to simplify expressions involving the sine of a sum of two angles. The formula is given by:

$\sin(x+y) = \sin x \cos y + \cos x \sin y$

This formula can be used to find the sine of a sum of two angles by breaking it down into the product of the sines and cosines of the individual angles.

Option A: $x = \frac{\pi}{3}, y = \frac{\pi}{4}$

Let's start by analyzing the first option, which states that $x = \frac{\pi}{3}$ and $y = \frac{\pi}{4}$ . We can use the sum formula for sine to find the expression for $\sin \frac{7 \pi}{12}$ in this case.

$\sin \frac{7 \pi}{12} = \sin \left(\frac{\pi}{3} + \frac{\pi}{4}\right)$

Using the sum formula for sine, we can rewrite this expression as:

$\sin \frac{7 \pi}{12} = \sin \frac{\pi}{3} \cos \frac{\pi}{4} + \cos \frac{\pi}{3} \sin \frac{\pi}{4}$

Now, we can use the values of the sine and cosine functions for $\frac{\pi}{3}$ and $\frac{\pi}{4}$ to simplify this expression.

$\sin \frac{7 \pi}{12} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Simplifying this expression, we get:

$\sin \frac{7 \pi}{12} = \frac{\sqrt{6} + \sqrt{2}}{4}$

Option B: $x = \frac{5 \pi}{6}, y = -\frac{\pi}{4}$

Now, let's analyze the second option, which states that $x = \frac{5 \pi}{6}$ and $y = -\frac{\pi}{4}$ . We can use the sum formula for sine to find the expression for $\sin \frac{7 \pi}{12}$ in this case.

$\sin \frac{7 \pi}{12} = \sin \left(\frac{5 \pi}{6} - \frac{\pi}{4}\right)$

Using the sum formula for sine, we can rewrite this expression as:

$\sin \frac{7 \pi}{12} = \sin \frac{5 \pi}{6} \cos \left(-\frac{\pi}{4}\right) + \cos \frac{5 \pi}{6} \sin \left(-\frac{\pi}{4}\right)$

Now, we can use the values of the sine and cosine functions for $\frac{5 \pi}{6}$ and $-\frac{\pi}{4}$ to simplify this expression.

$\sin \frac{7 \pi}{12} = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$

Simplifying this expression, we get:

$\sin \frac{7 \pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$

Option C: $x = \frac{3 \pi}{4}, y = \frac{\pi}{3}$

Finally, let's analyze the third option, which states that $x = \frac{3 \pi}{4}$ and $y = \frac{\pi}{3}$ . We can use the sum formula for sine to find the expression for $\sin \frac{7 \pi}{12}$ in this case.

$\sin \frac{7 \pi}{12} = \sin \left(\frac{3 \pi}{4} + \frac{\pi}{3}\right)$

Using the sum formula for sine, we can rewrite this expression as:

$\sin \frac{7 \pi}{12} = \sin \frac{3 \pi}{4} \cos \frac{\pi}{3} + \cos \frac{3 \pi}{4} \sin \frac{\pi}{3}$

Now, we can use the values of the sine and cosine functions for $\frac{3 \pi}{4}$ and $\frac{\pi}{3}$ to simplify this expression.

$\sin \frac{7 \pi}{12} = \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \left(-\frac{\sqrt{2}}{2}\right) \cdot \frac{\sqrt{3}}{2}$

Simplifying this expression, we get:

$\sin \frac{7 \pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$

Conclusion

In this article, we have explored how to find the correct expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ using the given options. We have analyzed each option and used the sum formula for sine to simplify the expression. The results are as follows:

Option A: $\sin \frac{7 \pi}{12} = \frac{\sqrt{6} + \sqrt{2}}{4}$
Option B: $\sin \frac{7 \pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$
Option C: $\sin \frac{7 \pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$

$Therefore, the correct expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ is $\sin \left(\frac{5 \pi}{6} - \frac{\pi}{4}\right)$ or $\sin \left(\frac{3 \pi}{4} + \frac{\pi}{3}\right)$.<br/> **Frequently Asked Questions (FAQs) about Finding the Correct Expression for $\sin \frac{7 \pi}{12}$ in the Form $\sin(x+y)$**$

Q: What is the sum formula for sine?

A: The sum formula for sine is a fundamental concept in trigonometry that allows us to simplify expressions involving the sine of a sum of two angles. The formula is given by:

$\sin(x+y) = \sin x \cos y + \cos x \sin y$

Q: How do I use the sum formula for sine to find the expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ ?

A: To use the sum formula for sine to find the expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ , you need to identify the values of $x$ and $y$ that satisfy the equation. In this case, we have three options:

Option A: $x = \frac{\pi}{3}, y = \frac{\pi}{4}$
Option B: $x = \frac{5 \pi}{6}, y = -\frac{\pi}{4}$
Option C: $x = \frac{3 \pi}{4}, y = \frac{\pi}{3}$

You can use the sum formula for sine to simplify the expression for each option and find the correct expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ .

Q: What are the results for each option?

A: The results for each option are as follows:

Option A: $\sin \frac{7 \pi}{12} = \frac{\sqrt{6} + \sqrt{2}}{4}$
Option B: $\sin \frac{7 \pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$
Option C: $\sin \frac{7 \pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$

Q: Which option is the correct expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ ?

A: The correct expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ is $\sin \left(\frac{5 \pi}{6} - \frac{\pi}{4}\right)$ or $\sin \left(\frac{3 \pi}{4} + \frac{\pi}{3}\right)$ .

Q: How do I simplify the expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ ?

A: To simplify the expression for $\sin \frac{7 \pi}{12}$ in the form $\sin(x+y)$ , you can use the sum formula for sine and the values of the sine and cosine functions for the given angles. You can also use algebraic manipulations to simplify the expression.

Q: What are the key concepts and formulas used in this article?

A: The key concepts and formulas used in this article are:

The sum formula for sine: $\sin(x+y) = \sin x \cos y + \cos x \sin y$
The values of the sine and cosine functions for the given angles
Algebraic manipulations to simplify the expression

Q: How can I apply the concepts and formulas used in this article to other problems?

A: You can apply the concepts and formulas used in this article to other problems by:

Using the sum formula for sine to simplify expressions involving the sine of a sum of two angles
Identifying the values of $x$ and $y$ that satisfy the equation
Using algebraic manipulations to simplify the expression
Applying the concepts and formulas to other trigonometric functions, such as cosine and tangent.