Understanding Addition and Subtraction Formulas

Oct 13, 2025 by miodeya.com 43 views

Introduction

Trigonometric functions, such as sine, cosine, and tangent, are essential in mathematics and have numerous applications in various fields, including physics, engineering, and navigation. One of the fundamental concepts in trigonometry is the use of addition and subtraction formulas to find the exact values of trigonometric expressions. In this article, we will explore how to use these formulas to find the exact value of the expression $\sin \left(-\frac{7 \pi}{12}\right)$ .

Understanding Addition and Subtraction Formulas

Before we dive into the solution, it's essential to understand the addition and subtraction formulas for sine, cosine, and tangent functions. 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

Similarly, the addition and subtraction formulas for cosine are:

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

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

And the addition and subtraction formulas for tangent are:

\tan (a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}

\tan (a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}

Finding the Exact Value of $\sin \left(-\frac{7 \pi}{12}\right)$

To find the exact value of $\sin \left(-\frac{7 \pi}{12}\right)$ , we can use the subtraction formula for sine:

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

In this case, we have $a = -\frac{7 \pi}{12}$ and $b = \frac{\pi}{3}$ . We can rewrite the expression as:

\sin \left(-\frac{7 \pi}{12} - \frac{\pi}{3}\right)

Using the subtraction formula, we get:

\sin \left(-\frac{7 \pi}{12} - \frac{\pi}{3}\right) = \sin \left(-\frac{7 \pi}{12}\right) \cos \left(\frac{\pi}{3}\right) - \cos \left(-\frac{7 \pi}{12}\right) \sin \left(\frac{\pi}{3}\right)

Evaluating the Trigonometric Functions

To evaluate the trigonometric functions, we need to find the values of $\sin \left(-\frac{7 \pi}{12}\right)$ , $\cos \left(-\frac{7 \pi}{12}\right)$ , $\cos \left(\frac{\pi}{3}\right)$ , and $\sin \left(\frac{\pi}{3}\right)$ . We can use the unit circle or reference angles to find these values.

Using the unit circle, we can find that:

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

\cos \left(-\frac{7 \pi}{12}\right) = \frac{1}{2}

\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}

\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}

Substituting the Values

Now that we have the values of the trigonometric functions, we can substitute them into the expression:

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

Simplifying the Expression

To simplify the expression, we can combine like terms:

\sin \left(-\frac{7 \pi}{12} - \frac{\pi}{3}\right) = -\frac{\sqrt{3}}{4} - \frac{\sqrt{3}}{4}

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

Conclusion

In this article, we used the subtraction formula for sine to find the exact value of $\sin \left(-\frac{7 \pi}{12}\right)$ . We evaluated the trigonometric functions using the unit circle and reference angles, and then substituted the values into the expression. Finally, we simplified the expression to find the exact value of $\sin \left(-\frac{7 \pi}{12}\right)$ . This example demonstrates the importance of using addition and subtraction formulas to find exact values of trigonometric expressions.

Applications of Trigonometry

Trigonometry has numerous applications in various fields, including physics, engineering, and navigation. Some of the applications of trigonometry include:

Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
Engineering: Trigonometry is used to design and build structures, such as bridges and buildings.
Navigation: Trigonometry is used to determine the position and direction of objects, such as ships and aircraft.

Real-World Examples

Here are some real-world examples of the use of trigonometry:

Building Design: Architects use trigonometry to design buildings and ensure that they are structurally sound.
GPS Navigation: GPS navigation systems use trigonometry to determine the position and direction of vehicles.
Medical Imaging: Medical imaging techniques, such as MRI and CT scans, use trigonometry to create detailed images of the body.

Final Thoughts

In conclusion, trigonometry is a fundamental concept in mathematics that has numerous applications in various fields. The use of addition and subtraction formulas is essential in finding exact values of trigonometric expressions. By understanding and applying these formulas, we can solve a wide range of problems in physics, engineering, and navigation.

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some of the most frequently asked questions about trigonometry.

Q1: What is Trigonometry?

A1: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of triangles, particularly right triangles, and the relationships between their sides and angles.

Q2: What are the Basic Trigonometric Functions?

A2: The basic trigonometric functions are sine, cosine, and tangent. These functions are used to describe the relationships between the sides and angles of triangles.

Q3: What is the Difference Between Sine, Cosine, and Tangent?

A3: Sine, cosine, and tangent are three basic trigonometric functions that describe the relationships between the sides and angles of triangles. Sine is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Cosine is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Tangent is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q4: How Do I Use Trigonometry in Real-Life Situations?

A4: Trigonometry has numerous applications in various fields, including physics, engineering, and navigation. Some examples of real-life situations where trigonometry is used include:

Building Design: Architects use trigonometry to design buildings and ensure that they are structurally sound.
GPS Navigation: GPS navigation systems use trigonometry to determine the position and direction of vehicles.
Medical Imaging: Medical imaging techniques, such as MRI and CT scans, use trigonometry to create detailed images of the body.

Q5: What is the Unit Circle?

A5: The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. It is used to define the values of trigonometric functions for angles.

Q6: How Do I Use the Unit Circle to Find Trigonometric Values?

A6: The unit circle is used to find the values of trigonometric functions for angles. To use the unit circle, you need to know the coordinates of the point on the circle that corresponds to the angle you are interested in.

Q7: What is the Difference Between a Right Triangle and an Oblique Triangle?

A7: A right triangle is a triangle with one right angle (90 degrees). An oblique triangle is a triangle with no right angles.

Q8: How Do I Use Trigonometry to Solve Problems Involving Right Triangles?

A8: To use trigonometry to solve problems involving right triangles, you need to know the lengths of the sides of the triangle and the measure of one of the angles. You can then use the trigonometric functions to find the lengths of the other sides and the measure of the other angles.

Q9: What is the Law of Sines?

A9: The Law of Sines is a formula that relates the lengths of the sides of a triangle to the sines of its angles. It is used to solve problems involving triangles.

Q10: How Do I Use the Law of Sines to Solve Problems Involving Triangles?

A10: To use the Law of Sines to solve problems involving triangles, you need to know the lengths of the sides of the triangle and the measure of one of the angles. You can then use the Law of Sines to find the lengths of the other sides and the measure of the other angles.

Conclusion

In this article, we have answered some of the most frequently asked questions about trigonometry. We have covered topics such as the basic trigonometric functions, the unit circle, and the Law of Sines. We hope that this article has been helpful in answering your questions and providing you with a better understanding of trigonometry.

Additional Resources

If you are interested in learning more about trigonometry, there are many additional resources available. Some of these resources include:

Textbooks: There are many textbooks available that cover trigonometry in detail.
Online Resources: There are many online resources available that provide tutorials, examples, and practice problems for trigonometry.
Video Lectures: There are many video lectures available that cover trigonometry in detail.
Practice Problems: There are many practice problems available that can help you to practice and improve your skills in trigonometry.

Final Thoughts

In conclusion, trigonometry is a fundamental concept in mathematics that has numerous applications in various fields. It is used to describe the relationships between the sides and angles of triangles and has numerous applications in physics, engineering, and navigation. We hope that this article has been helpful in answering your questions and providing you with a better understanding of trigonometry.