Use An Addition Or Subtraction Formula To Write The Expression As A Trigonometric Function Of One Number.$\cos \left(\frac{4 \pi}{7}\right) \cos \left(\frac{5 \pi}{21}\right) + \sin \left(\frac{4 \pi}{7}\right) \sin \left(\frac{5

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore the use of addition and subtraction formulas in trigonometry to simplify complex trigonometric expressions.

Addition and Subtraction Formulas

The addition and subtraction formulas are used to simplify complex trigonometric expressions by expressing them in terms of a single trigonometric function. The addition formula for cosine is given by:

$\cos (a + b) = \cos a \cos b - \sin a \sin b$

The subtraction formula for cosine is given by:

$\cos (a - b) = \cos a \cos b + \sin a \sin b$

The addition formula for sine is given by:

$\sin (a + b) = \sin a \cos b + \cos a \sin b$

The subtraction formula for sine is given by:

$\sin (a - b) = \sin a \cos b - \cos a \sin b$

Using Addition and Subtraction Formulas to Simplify Expressions

Now, let's use the addition and subtraction formulas to simplify the given expression:

$\cos \left(\frac{4 \pi}{7}\right) \cos \left(\frac{5 \pi}{21}\right) + \sin \left(\frac{4 \pi}{7}\right) \sin \left(\frac{5 \pi}{21}\right)$

We can use the addition formula for cosine to simplify the expression:

$\cos \left(\frac{4 \pi}{7}\right) \cos \left(\frac{5 \pi}{21}\right) + \sin \left(\frac{4 \pi}{7}\right) \sin \left(\frac{5 \pi}{21}\right) = \cos \left(\frac{4 \pi}{7} + \frac{5 \pi}{21}\right)$

Using the addition formula for cosine, we get:

$\cos \left(\frac{4 \pi}{7} + \frac{5 \pi}{21}\right) = \cos \left(\frac{28 \pi}{21} + \frac{5 \pi}{21}\right) = \cos \left(\frac{33 \pi}{21}\right)$

Simplifying the Expression Further

We can simplify the expression further by using the fact that the cosine function has a period of $2 \pi$. We can rewrite the expression as:

$\cos \left(\frac{33 \pi}{21}\right) = \cos \left(\frac{33 \pi}{21} - 2 \pi\right)$

Using the subtraction formula for cosine, we get:

$\cos \left(\frac{33 \pi}{21} - 2 \pi\right) = \cos \left(\frac{33 \pi}{21}\right) \cos (2 \pi) + \sin \left(\frac{33 \pi}{21}\right) \sin (2 \pi)$

Since $\cos (2 \pi) = 1$ and $\sin (2 \pi) = 0$, we get:

$\cos \left(\frac{33 \pi}{21}\right) = \cos \left(\frac{33 \pi}{21}\right)$

Conclusion

In this article, we used the addition and subtraction formulas to simplify a complex trigonometric expression. We showed that the expression can be simplified by using the addition formula for cosine and then simplifying the resulting expression further by using the fact that the cosine function has a period of $2 \pi$. This demonstrates the power of using addition and subtraction formulas in trigonometry to simplify complex expressions.

Applications of Addition and Subtraction Formulas

Addition and subtraction formulas have numerous applications in various fields, including physics, engineering, and navigation. They are used to simplify complex trigonometric expressions that arise in the study of periodic phenomena, such as sound waves, light waves, and electrical signals. They are also used in the design of electronic circuits, such as filters and amplifiers, where complex trigonometric expressions are used to describe the behavior of the circuit.

Example Problems

1. Simplify the expression $\cos \left(\frac{3 \pi}{5}\right) \cos \left(\frac{2 \pi}{7}\right) + \sin \left(\frac{3 \pi}{5}\right) \sin \left(\frac{2 \pi}{7}\right)$ using the addition formula for cosine.
2. Simplify the expression $\sin \left(\frac{4 \pi}{9}\right) \cos \left(\frac{3 \pi}{11}\right) + \cos \left(\frac{4 \pi}{9}\right) \sin \left(\frac{3 \pi}{11}\right)$ using the addition formula for sine.
3. Simplify the expression $\cos \left(\frac{5 \pi}{13}\right) \cos \left(\frac{2 \pi}{17}\right) - \sin \left(\frac{5 \pi}{13}\right) \sin \left(\frac{2 \pi}{17}\right)$ using the subtraction formula for cosine.

Practice Problems

1. Simplify the expression $\cos \left(\frac{2 \pi}{3}\right) \cos \left(\frac{4 \pi}{15}\right) + \sin \left(\frac{2 \pi}{3}\right) \sin \left(\frac{4 \pi}{15}\right)$ using the addition formula for cosine.
2. Simplify the expression $\sin \left(\frac{3 \pi}{8}\right) \cos \left(\frac{2 \pi}{13}\right) - \cos \left(\frac{3 \pi}{8}\right) \sin \left(\frac{2 \pi}{13}\right)$ using the subtraction formula for sine.
3. Simplify the expression $\cos \left(\frac{7 \pi}{11}\right) \cos \left(\frac{3 \pi}{19}\right) + \sin \left(\frac{7 \pi}{11}\right) \sin \left(\frac{3 \pi}{19}\right)$ using the addition formula for cosine.

Glossary of Terms

• Addition formula for cosine: $\cos (a + b) = \cos a \cos b - \sin a \sin b$
• Subtraction formula for cosine: $\cos (a - b) = \cos a \cos b + \sin a \sin b$
• Addition formula for sine: $\sin (a + b) = \sin a \cos b + \cos a \sin b$
• Subtraction formula for sine: $\sin (a - b) = \sin a \cos b - \cos a \sin b$
• Periodic function: A function that repeats itself at regular intervals.
• Trigonometric function: A function that relates the angles of a triangle to the ratios of the lengths of its sides.