Understanding Angle Sum and Difference Formulas

Oct 13, 2025 by wbrum.org 53 views

Introduction

Trigonometric identities are a fundamental concept in mathematics, and they play a crucial role in solving various problems in mathematics and physics. One of the most important trigonometric identities is the angle sum and difference formulas, which are used to find the exact value of trigonometric functions. In this article, we will discuss how to use the appropriate angle sum or difference formula to find the exact value of $\cos(-15^\circ)$ .

Understanding Angle Sum and Difference Formulas

The angle sum and difference formulas are used to find the exact value of trigonometric functions. The angle sum formula is given by:

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

The angle difference formula is given by:

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

Choosing the Right Formula

To find the exact value of $\cos(-15^\circ)$ , we need to choose the right formula. Since we are dealing with a negative angle, we will use the angle difference formula.

Applying the Angle Difference Formula

We can rewrite $\cos(-15^\circ)$ as $\cos(15^\circ - 180^\circ)$ , since $\cos(-\theta) = \cos(\theta - 180^\circ)$ . Now, we can apply the angle difference formula:

\cos(15^\circ - 180^\circ) = \cos 15^\circ \cos 180^\circ + \sin 15^\circ \sin 180^\circ

Evaluating the Trigonometric Functions

We know that $\cos 180^\circ = -1$ and $\sin 180^\circ = 0$ . Therefore, we can substitute these values into the equation:

\cos(15^\circ - 180^\circ) = \cos 15^\circ (-1) + \sin 15^\circ (0)

Simplifying the equation, we get:

\cos(15^\circ - 180^\circ) = -\cos 15^\circ

Finding the Exact Value

To find the exact value of $\cos 15^\circ$ , we can use the following identity:

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

Substituting this value into the equation, we get:

\cos(15^\circ - 180^\circ) = -\frac{\sqrt{2} + \sqrt{6}}{4}

Conclusion

In this article, we discussed how to use the appropriate angle sum or difference formula to find the exact value of $\cos(-15^\circ)$ . We chose the angle difference formula and applied it to the given problem. We also evaluated the trigonometric functions and found the exact value of $\cos 15^\circ$ . Finally, we substituted this value into the equation to find the exact value of $\cos(-15^\circ)$ . We hope that this article has provided a comprehensive guide to solving trigonometric identities.

Additional Resources

For more information on trigonometric identities, we recommend the following resources:

Frequently Asked Questions

Q: What is the angle sum formula? A: The angle sum formula is given by $\cos(A+B) = \cos A \cos B - \sin A \sin B$ .

Q: What is the angle difference formula? A: The angle difference formula is given by $\cos(A-B) = \cos A \cos B + \sin A \sin B$ .

Q: How do I choose the right formula? A: To choose the right formula, you need to determine whether you are dealing with an angle sum or angle difference problem.

Q: How do I apply the angle difference formula? A: To apply the angle difference formula, you need to rewrite the problem in the form $\cos(A-B)$ and then substitute the values of the trigonometric functions.

$Q: What are the values of $\cos 180^\circ$ and $\sin 180^\circ$? A: The values of $\cos 180^\circ$ and $\sin 180^\circ$ are $-1$ and $0$, respectively.<br/> **Trigonometric Identities: A Comprehensive Guide**$

Q&A: Trigonometric Identities

Q: What is the difference between the angle sum and difference formulas? A: The angle sum formula is used to find the exact value of $\cos(A+B)$ , while the angle difference formula is used to find the exact value of $\cos(A-B)$ .

Q: How do I choose the right formula? A: To choose the right formula, you need to determine whether you are dealing with an angle sum or angle difference problem. If you are adding two angles, use the angle sum formula. If you are subtracting two angles, use the angle difference formula.

Q: What is the angle sum formula? A: The angle sum formula is given by $\cos(A+B) = \cos A \cos B - \sin A \sin B$ .

Q: What is the angle difference formula? A: The angle difference formula is given by $\cos(A-B) = \cos A \cos B + \sin A \sin B$ .

Q: How do I apply the angle difference formula? A: To apply the angle difference formula, you need to rewrite the problem in the form $\cos(A-B)$ and then substitute the values of the trigonometric functions.

Q: What are the values of $\cos 180^\circ$ and $\sin 180^\circ$ ? A: The values of $\cos 180^\circ$ and $\sin 180^\circ$ are $-1$ and $0$ , respectively.

Q: How do I find the exact value of $\cos(-15^\circ)$ ? A: To find the exact value of $\cos(-15^\circ)$ , you can use the angle difference formula. First, rewrite $\cos(-15^\circ)$ as $\cos(15^\circ - 180^\circ)$ . Then, apply the angle difference formula and substitute the values of the trigonometric functions.

Q: What is the exact value of $\cos 15^\circ$ ? A: The exact value of $\cos 15^\circ$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

Q: How do I simplify the equation $\cos(15^\circ - 180^\circ) = -\cos 15^\circ$ ? A: To simplify the equation, substitute the value of $\cos 15^\circ$ into the equation and simplify.

Q: What is the final answer for $\cos(-15^\circ)$ ? A: The final answer for $\cos(-15^\circ)$ is $-\frac{\sqrt{2} + \sqrt{6}}{4}$ .

Q: What are some common trigonometric identities? A: Some common trigonometric identities include:

$\sin^2 A + \cos^2 A = 1$
$\cos(A+B) = \cos A \cos B - \sin A \sin B$
$\cos(A-B) = \cos A \cos B + \sin A \sin B$
$\sin(A+B) = \sin A \cos B + \cos A \sin B$
$\sin(A-B) = \sin A \cos B - \cos A \sin B$

Q: How do I use trigonometric identities to solve problems? A: To use trigonometric identities to solve problems, first identify the type of problem you are dealing with. Then, choose the appropriate trigonometric identity and apply it to the problem. Finally, simplify the equation and solve for the unknown value.

Q: What are some tips for solving trigonometric identities? A: Some tips for solving trigonometric identities include:

Read the problem carefully and identify the type of problem you are dealing with.
Choose the appropriate trigonometric identity and apply it to the problem.
Simplify the equation and solve for the unknown value.
Check your work by plugging the solution back into the original equation.

Q: What are some common mistakes to avoid when solving trigonometric identities? A: Some common mistakes to avoid when solving trigonometric identities include:

Not reading the problem carefully and identifying the type of problem you are dealing with.
Choosing the wrong trigonometric identity and applying it to the problem.
Not simplifying the equation and solving for the unknown value.
Not checking your work by plugging the solution back into the original equation.