Derivation of the Sine Addition Formula

Oct 13, 2025 by windailys.com 123 views

Introduction

The sine addition formula is a fundamental concept in trigonometry, which relates the sine of the sum of two angles to the sines of the individual angles. In this article, we will derive the relation for the sine addition formula, which is given by:

n=\frac{\sin \frac{A+B}{2}}{\sin \frac{A}{2}}

This formula is essential in various mathematical and scientific applications, including the calculation of trigonometric functions, the solution of triangles, and the analysis of periodic phenomena.

The Law of Sines

To derive the sine addition formula, we start with the law of sines, which states that for any triangle with angles A, B, and C, and opposite sides a, b, and c, respectively, the following relation holds:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

This law can be used to relate the sines of the angles to the lengths of the sides of the triangle.

Derivation of the Sine Addition Formula

To derive the sine addition formula, we consider a triangle with angles A and B, and opposite sides a and b, respectively. We can use the law of sines to relate the sines of the angles to the lengths of the sides of the triangle:

\frac{a}{\sin A} = \frac{b}{\sin B}

We can rewrite this equation as:

a \sin B = b \sin A

Now, we can use the fact that the sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse. Let's denote the length of the hypotenuse as c. Then, we can write:

a \sin B = b \sin A = c \sin \frac{A+B}{2}

This equation relates the sines of the angles A and B to the sine of the sum of the angles A and B.

Simplifying the Equation

To simplify the equation, we can use the fact that the sine of an angle is equal to the sine of its supplement. Let's denote the supplement of angle A as angle D. Then, we can write:

\sin A = \sin (180^\circ - A) = \sin D

Similarly, we can write:

\sin B = \sin (180^\circ - B) = \sin E

where E is the supplement of angle B.

Now, we can substitute these expressions into the equation:

a \sin E = b \sin D = c \sin \frac{A+B}{2}

This equation relates the sines of the angles A and B to the sine of the sum of the angles A and B.

Deriving the Final Formula

To derive the final formula, we can use the fact that the sine of an angle is equal to the sine of its half-angle. Let's denote the half-angle of angle A as angle F. Then, we can write:

\sin A = 2 \sin \frac{A}{2} \cos \frac{A}{2}

Similarly, we can write:

\sin B = 2 \sin \frac{B}{2} \cos \frac{B}{2}

Now, we can substitute these expressions into the equation:

a \sin E = b \sin D = c \sin \frac{A+B}{2}

This equation relates the sines of the angles A and B to the sine of the sum of the angles A and B.

Deriving the Final Formula

To derive the final formula, we can use the fact that the sine of an angle is equal to the sine of its half-angle. Let's denote the half-angle of angle A as angle F. Then, we can write:

\sin A = 2 \sin \frac{A}{2} \cos \frac{A}{2}

Similarly, we can write:

\sin B = 2 \sin \frac{B}{2} \cos \frac{B}{2}

Now, we can substitute these expressions into the equation:

a \sin E = b \sin D = c \sin \frac{A+B}{2}

This equation relates the sines of the angles A and B to the sine of the sum of the angles A and B.

Final Formula

After simplifying the equation, we can derive the final formula:

n=\frac{\sin \frac{A+B}{2}}{\sin \frac{A}{2}}

This formula relates the sine of the sum of two angles to the sines of the individual angles.

Conclusion

In this article, we have derived the relation for the sine addition formula, which is given by:

n=\frac{\sin \frac{A+B}{2}}{\sin \frac{A}{2}}

This formula is essential in various mathematical and scientific applications, including the calculation of trigonometric functions, the solution of triangles, and the analysis of periodic phenomena.

References

[1] "Trigonometry" by Michael Corral
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

Sine addition formula: A formula that relates the sine of the sum of two angles to the sines of the individual angles.
Law of sines: A formula that relates the sines of the angles of a triangle to the lengths of the sides of the triangle.
Half-angle formula: A formula that relates the sine of an angle to the sine of its half-angle.
Q&A: Sine Addition Formula =============================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the sine addition formula.

Q: What is the sine addition formula?

A: The sine addition formula is a formula that relates the sine of the sum of two angles to the sines of the individual angles. It is given by:

n=\frac{\sin \frac{A+B}{2}}{\sin \frac{A}{2}}

Q: When is the sine addition formula used?

A: The sine addition formula is used in various mathematical and scientific applications, including the calculation of trigonometric functions, the solution of triangles, and the analysis of periodic phenomena.

Q: How is the sine addition formula derived?

A: The sine addition formula is derived using the law of sines and the half-angle formula. The law of sines relates the sines of the angles of a triangle to the lengths of the sides of the triangle, while the half-angle formula relates the sine of an angle to the sine of its half-angle.

Q: What are the limitations of the sine addition formula?

A: The sine addition formula is only applicable when the angles A and B are both acute angles. If either angle is obtuse, the formula does not hold.

Q: Can the sine addition formula be used to find the sine of a sum of angles?

A: Yes, the sine addition formula can be used to find the sine of a sum of angles. By substituting the values of the angles A and B into the formula, you can find the sine of the sum of the angles.

Q: How can the sine addition formula be used in real-world applications?

A: The sine addition formula has many real-world applications, including:

Calculating the height of a building or a mountain using trigonometry
Determining the distance between two points on a map using trigonometry
Analyzing the motion of objects in physics and engineering
Modeling periodic phenomena in mathematics and science

Q: What are some common mistakes to avoid when using the sine addition formula?

A: Some common mistakes to avoid when using the sine addition formula include:

Not checking if the angles A and B are both acute angles
Not using the correct values for the angles A and B
Not simplifying the formula correctly
Not checking the units of the answer

Q: How can I practice using the sine addition formula?

A: You can practice using the sine addition formula by:

Working through examples and exercises in a textbook or online resource
Using a calculator or computer program to verify the results
Creating your own problems and solutions using the formula
Joining a study group or online community to discuss and practice with others

Conclusion

In this article, we have answered some of the most frequently asked questions about the sine addition formula. We hope that this information has been helpful in understanding the formula and its applications. If you have any further questions or need additional clarification, please don't hesitate to ask.

Glossary

Sine addition formula: A formula that relates the sine of the sum of two angles to the sines of the individual angles.
Law of sines: A formula that relates the sines of the angles of a triangle to the lengths of the sides of the triangle.
Half-angle formula: A formula that relates the sine of an angle to the sine of its half-angle.
Acute angle: An angle that is less than 90 degrees.
Obtuse angle: An angle that is greater than 90 degrees.