Sine and Cosine Addition Formulae

Oct 13, 2025 by windailys.com 37 views

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 delve into a complex trigonometric expression and simplify it using various trigonometric identities.

The Expression

The given expression is:

\frac{\sin \frac{\pi}{10} \cdot \sin \frac{\pi}{5} + \cos \frac{\pi}{5} \cdot \cos \frac{\pi}{10}}{\sin \frac{\pi}{5} \cdot \sin \frac{2 \pi}{15} - \cos \frac{\pi}{5} \cdot \sin \frac{11 \pi}{30}}

This expression appears to be a complex combination of sine and cosine functions. To simplify it, we need to use various trigonometric identities and formulas.

Trigonometric Identities

Before we proceed with simplifying the expression, let's recall some basic trigonometric identities:

Sine and Cosine Addition Formulae: $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and $\cos (A + B) = \cos A \cos B - \sin A \sin B$
Sine and Cosine Difference Formulae: $\sin (A - B) = \sin A \cos B - \cos A \sin B$ and $\cos (A - B) = \cos A \cos B + \sin A \sin B$
Double Angle Formulae: $\sin 2A = 2 \sin A \cos A$ and $\cos 2A = 1 - 2 \sin^2 A = 2 \cos^2 A - 1$
Product-to-Sum Formulae: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$ and $\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$

Simplifying the Expression

Now that we have recalled some basic trigonometric identities, let's simplify the given expression.

Numerator

The numerator of the expression is:

\sin \frac{\pi}{10} \cdot \sin \frac{\pi}{5} + \cos \frac{\pi}{5} \cdot \cos \frac{\pi}{10}

Using the product-to-sum formulae, we can rewrite the numerator as:

\frac{1}{2}[\cos(\frac{\pi}{10} - \frac{\pi}{5}) + \cos(\frac{\pi}{10} + \frac{\pi}{5})]

Simplifying further, we get:

\frac{1}{2}[\cos(-\frac{\pi}{10}) + \cos(\frac{3\pi}{10})]

Using the cosine addition formula, we can rewrite the numerator as:

\frac{1}{2}[\cos(\frac{\pi}{10}) + \cos(\frac{3\pi}{10})]

Denominator

The denominator of the expression is:

\sin \frac{\pi}{5} \cdot \sin \frac{2 \pi}{15} - \cos \frac{\pi}{5} \cdot \sin \frac{11 \pi}{30}

Using the product-to-sum formulae, we can rewrite the denominator as:

\frac{1}{2}[\cos(\frac{\pi}{5} - \frac{2\pi}{15}) - \cos(\frac{\pi}{5} + \frac{11\pi}{30})]

Simplifying further, we get:

\frac{1}{2}[\cos(\frac{\pi}{15}) - \cos(\frac{17\pi}{30})]

Using the cosine addition formula, we can rewrite the denominator as:

\frac{1}{2}[\cos(\frac{\pi}{15}) - \cos(\frac{\pi}{2} - \frac{\pi}{30})]

Final Simplification

Now that we have simplified the numerator and denominator, let's combine them to get the final simplified expression.

\frac{\frac{1}{2}[\cos(\frac{\pi}{10}) + \cos(\frac{3\pi}{10})]}{\frac{1}{2}[\cos(\frac{\pi}{15}) - \cos(\frac{\pi}{2} - \frac{\pi}{30})]}

Simplifying further, we get:

\frac{\cos(\frac{\pi}{10}) + \cos(\frac{3\pi}{10})}{\cos(\frac{\pi}{15}) - \cos(\frac{\pi}{2} - \frac{\pi}{30})}

Using the cosine difference formula, we can rewrite the denominator as:

\cos(\frac{\pi}{15}) - \cos(\frac{\pi}{30})

Simplifying further, we get:

\frac{\cos(\frac{\pi}{10}) + \cos(\frac{3\pi}{10})}{\cos(\frac{\pi}{15}) - \cos(\frac{\pi}{30})}

Using the cosine addition formula, we can rewrite the numerator as:

\cos(\frac{\pi}{10}) + \cos(\frac{3\pi}{10}) = 2\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})

Simplifying further, we get:

\frac{2\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\cos(\frac{\pi}{15}) - \cos(\frac{\pi}{30})}

Using the cosine difference formula, we can rewrite the denominator as:

\cos(\frac{\pi}{15}) - \cos(\frac{\pi}{30}) = 2\sin(\frac{\pi}{30})\sin(\frac{\pi}{15})

Simplifying further, we get:

\frac{2\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{2\sin(\frac{\pi}{30})\sin(\frac{\pi}{15})}

Cancelling out the common factor of 2, we get:

\frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\sin(\frac{\pi}{30})\sin(\frac{\pi}{15})}

Using the product-to-sum formulae, we can rewrite the expression as:

\frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\sin(\frac{\pi}{30})\sin(\frac{\pi}{15})} = \frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\frac{1}{2}[\cos(\frac{\pi}{30} - \frac{\pi}{15}) - \cos(\frac{\pi}{30} + \frac{\pi}{15})]}

Simplifying further, we get:

\frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\frac{1}{2}[\cos(\frac{\pi}{45}) - \cos(\frac{\pi}{15})]}

Using the cosine difference formula, we can rewrite the denominator as:

\frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\frac{1}{2}[\cos(\frac{\pi}{45}) - \cos(\frac{\pi}{15})]} = \frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\frac{1}{2}[\cos(\frac{\pi}{45}) - \cos(\frac{\pi}{15})]}

Simplifying further, we get:

\frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\frac{1}{2}[\cos(\frac{\pi}{45}) - \cos(\frac{\pi}{15})]} = \frac{2\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\cos(\frac{\pi}{45}) - \cos(\frac{\pi}{15})}

Using the cosine addition formula, we can rewrite the numerator as:

2\cos(\frac{\pi}{10})\cos(\frac{\pi}{5}) = \cos(\frac{\pi}{10} + \frac{\pi}{5}) + \cos(\frac{\pi}{10} - \frac{\pi}{5})

Simplifying further, we get:

\cos(\frac{\pi}{10} + \frac{\pi}{5}) + \cos(\frac{\pi}{10} - \frac{\pi}{5}) = \cos(\frac{3\pi}{10}) + \cos(\frac{\pi}{10})

Using the cosine difference formula, we can rewrite the denominator as:

\cos(\frac{\pi}{45}) - \cos(\frac{\pi}{15}) = 2\sin(\frac{\pi}{45})\sin(\frac{\pi}{15})

Simplifying further, we

Introduction

In our previous article, we simplified a complex trigonometric expression using various trigonometric identities and formulas. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q: What is the significance of the given expression?

A: The given expression is a complex trigonometric expression that appears to be a combination of sine and cosine functions. It is a fundamental expression in trigonometry and has numerous applications in various fields, including physics, engineering, and navigation.

Q: What are the key trigonometric identities used to simplify the expression?

A: The key trigonometric identities used to simplify the expression are:

Sine and Cosine Addition Formulae: $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and $\cos (A + B) = \cos A \cos B - \sin A \sin B$
Sine and Cosine Difference Formulae: $\sin (A - B) = \sin A \cos B - \cos A \sin B$ and $\cos (A - B) = \cos A \cos B + \sin A \sin B$
Double Angle Formulae: $\sin 2A = 2 \sin A \cos A$ and $\cos 2A = 1 - 2 \sin^2 A = 2 \cos^2 A - 1$
Product-to-Sum Formulae: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$ and $\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$

Q: How do you simplify the numerator of the expression?

A: To simplify the numerator, we use the product-to-sum formulae to rewrite the expression as:

\frac{1}{2}[\cos(\frac{\pi}{10} - \frac{\pi}{5}) + \cos(\frac{\pi}{10} + \frac{\pi}{5})]

Simplifying further, we get:

\frac{1}{2}[\cos(-\frac{\pi}{10}) + \cos(\frac{3\pi}{10})]

Using the cosine addition formula, we can rewrite the numerator as:

\frac{1}{2}[\cos(\frac{\pi}{10}) + \cos(\frac{3\pi}{10})]

Q: How do you simplify the denominator of the expression?

A: To simplify the denominator, we use the product-to-sum formulae to rewrite the expression as:

\frac{1}{2}[\cos(\frac{\pi}{5} - \frac{2\pi}{15}) - \cos(\frac{\pi}{5} + \frac{11\pi}{30})]

Simplifying further, we get:

\frac{1}{2}[\cos(\frac{\pi}{15}) - \cos(\frac{17\pi}{30})]

Using the cosine addition formula, we can rewrite the denominator as:

\frac{1}{2}[\cos(\frac{\pi}{15}) - \cos(\frac{\pi}{2} - \frac{\pi}{30})]

Q: What is the final simplified expression?

A: The final simplified expression is:

\frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\sin(\frac{\pi}{30})\sin(\frac{\pi}{15})}

Using the product-to-sum formulae, we can rewrite the expression as:

\frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\frac{1}{2}[\cos(\frac{\pi}{30} - \frac{\pi}{15}) - \cos(\frac{\pi}{30} + \frac{\pi}{15})]}

Simplifying further, we get:

\frac{\cos(\frac{\pi}{10})\cos(\frac{\pi}{5})}{\frac{1}{2}[\cos(\frac{\pi}{45}) - \cos(\frac{\pi}{15})]}

Q: What are the applications of the simplified expression?

A: The simplified expression has numerous applications in various fields, including:

Physics: The expression is used to describe the motion of objects in two-dimensional space.
Engineering: The expression is used to design and analyze mechanical systems, such as gears and linkages.
Navigation: The expression is used to calculate the position and velocity of objects in navigation systems.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of a complex trigonometric expression. We used various trigonometric identities and formulas to simplify the expression and provided examples of its applications in various fields.