The Sine Subtraction Formula

Oct 13, 2025 by miodeya.com 97 views

Introduction

Trigonometric functions are essential in mathematics, and understanding how to use addition and subtraction formulas is crucial for solving various problems. In this article, we will explore how to use these formulas to find the exact value of the expression $\sin \left(105^{\circ}\right)$ .

What are Addition and Subtraction Formulas?

Addition and subtraction formulas are used to simplify trigonometric expressions by combining different angles. These formulas are based on the relationships between the sine, cosine, and tangent functions.

The Sine Addition Formula

The sine addition formula states that:

\sin (A + B) = \sin A \cos B + \cos A \sin B

This formula allows us to find the sine of the sum of two angles by using the sines and cosines of the individual angles.

The Sine Subtraction Formula

The sine subtraction formula states that:

\sin (A - B) = \sin A \cos B - \cos A \sin B

This formula allows us to find the sine of the difference of two angles by using the sines and cosines of the individual angles.

Using the Sine Addition Formula to Find the Exact Value of $\sin \left(105^{\circ}\right)$

To find the exact value of $\sin \left(105^{\circ}\right)$ , we can use the sine addition formula. We can express $105^{\circ}$ as the sum of two angles: $90^{\circ}$ and $15^{\circ}$ .

\sin \left(105^{\circ}\right) = \sin \left(90^{\circ} + 15^{\circ}\right)

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

\sin \left(105^{\circ}\right) = \sin 90^{\circ} \cos 15^{\circ} + \cos 90^{\circ} \sin 15^{\circ}

We know that $\sin 90^{\circ} = 1$ and $\cos 90^{\circ} = 0$ , so the expression simplifies to:

\sin \left(105^{\circ}\right) = 1 \cdot \cos 15^{\circ} + 0 \cdot \sin 15^{\circ}

\sin \left(105^{\circ}\right) = \cos 15^{\circ}

Finding the Exact Value of $\cos 15^{\circ}$

To find the exact value of $\cos 15^{\circ}$ , we can use the cosine addition formula. We can express $15^{\circ}$ as the sum of two angles: $45^{\circ}$ and $30^{\circ}$ .

\cos 15^{\circ} = \cos \left(45^{\circ} - 30^{\circ}\right)

Using the cosine subtraction formula, we can rewrite this expression as:

\cos 15^{\circ} = \cos 45^{\circ} \cos 30^{\circ} + \sin 45^{\circ} \sin 30^{\circ}

We know that $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$ , $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ , $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ , and $\sin 30^{\circ} = \frac{1}{2}$ , so the expression simplifies to:

\cos 15^{\circ} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}

\cos 15^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4}

Conclusion

In this article, we used the sine addition and subtraction formulas to find the exact value of the expression $\sin \left(105^{\circ}\right)$ . We expressed $105^{\circ}$ as the sum of two angles, $90^{\circ}$ and $15^{\circ}$ , and used the sine addition formula to rewrite the expression. We then used the cosine addition formula to find the exact value of $\cos 15^{\circ}$ , which is $\frac{\sqrt{6} + \sqrt{2}}{4}$ . Therefore, the exact value of $\sin \left(105^{\circ}\right)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

Final Answer

$The final answer is $\boxed{\frac{\sqrt{6} + \sqrt{2}}{4}}$.<br/> **Q&A: Using Addition and Subtraction Formulas to Find Exact Values of Trigonometric Expressions**$

Introduction

In our previous article, we explored how to use addition and subtraction formulas to find the exact value of the expression $\sin \left(105^{\circ}\right)$ . In this article, we will answer some frequently asked questions about using these formulas to find exact values of trigonometric expressions.

Q: What are the most common trigonometric addition and subtraction formulas?

A: The most common trigonometric addition and subtraction formulas are:

Sine addition formula: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
Sine subtraction formula: $\sin (A - B) = \sin A \cos B - \cos A \sin B$
Cosine addition formula: $\cos (A + B) = \cos A \cos B - \sin A \sin B$
Cosine subtraction formula: $\cos (A - B) = \cos A \cos B + \sin A \sin B$

Q: How do I choose which formula to use?

A: To choose which formula to use, you need to determine whether you are adding or subtracting angles. If you are adding angles, use the addition formula. If you are subtracting angles, use the subtraction formula.

Q: What are some common angles that I can use to simplify trigonometric expressions?

A: Some common angles that you can use to simplify trigonometric expressions are:

$0^{\circ}$ : $\sin 0^{\circ} = 0$ , $\cos 0^{\circ} = 1$
$30^{\circ}$ : $\sin 30^{\circ} = \frac{1}{2}$ , $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$
$45^{\circ}$ : $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ , $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$
$60^{\circ}$ : $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$ , $\cos 60^{\circ} = \frac{1}{2}$
$90^{\circ}$ : $\sin 90^{\circ} = 1$ , $\cos 90^{\circ} = 0$

Q: How do I use the addition and subtraction formulas to find exact values of trigonometric expressions?

A: To use the addition and subtraction formulas to find exact values of trigonometric expressions, follow these steps:

Express the angle as the sum or difference of two angles.
Use the addition or subtraction formula to rewrite the expression.
Simplify the expression using the values of the trigonometric functions for the individual angles.

Q: What are some common mistakes to avoid when using the addition and subtraction formulas?

A: Some common mistakes to avoid when using the addition and subtraction formulas are:

Not using the correct formula for the problem.
Not simplifying the expression correctly.
Not using the values of the trigonometric functions for the individual angles.

Conclusion

In this article, we answered some frequently asked questions about using addition and subtraction formulas to find exact values of trigonometric expressions. We hope that this article has been helpful in clarifying some of the common mistakes and misconceptions about using these formulas. Remember to always use the correct formula for the problem, simplify the expression correctly, and use the values of the trigonometric functions for the individual angles.

Final Tips

Practice, practice, practice! The more you practice using the addition and subtraction formulas, the more comfortable you will become with them.
Use a calculator to check your answers and make sure you are getting the correct values.
Don't be afraid to ask for help if you are struggling with a problem.

Common Trigonometric Formulas

Here are some common trigonometric formulas that you may find useful:

Sine addition formula: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
Sine subtraction formula: $\sin (A - B) = \sin A \cos B - \cos A \sin B$
Cosine addition formula: $\cos (A + B) = \cos A \cos B - \sin A \sin B$
Cosine subtraction formula: $\cos (A - B) = \cos A \cos B + \sin A \sin B$

Common Trigonometric Values

Here are some common trigonometric values that you may find useful:

$\sin 0^{\circ} = 0$ , $\cos 0^{\circ} = 1$
$\sin 30^{\circ} = \frac{1}{2}$ , $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$
$\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ , $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$
$\sin 60^{\circ} = \frac{\sqrt{3}}{2}$ , $\cos 60^{\circ} = \frac{1}{2}$
$\sin 90^{\circ} = 1$ , $\cos 90^{\circ} = 0$