The Sine and Cosine Sum and Difference Identities

Oct 13, 2025 by miodeya.com 90 views

Introduction

Trigonometric identities are essential tools in mathematics, particularly in solving trigonometric expressions. The sum and difference identities are two fundamental identities that can be used to simplify and solve trigonometric expressions. In this article, we will focus on using the sine and cosine sum and difference identities to determine the value of a given expression.

The Sine and Cosine Sum and Difference Identities

The sine and cosine sum and difference identities are as follows:

Sine Sum Identity: $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$
Sine Difference Identity: $\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$
Cosine Sum Identity: $\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$
Cosine Difference Identity: $\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$

Using the Sine and Cosine Sum and Difference Identities

Now that we have the sine and cosine sum and difference identities, let's use them to solve the given expression.

The Given Expression

The given expression is:

$\cos\left(\frac{17\pi}{12}\right) = \cos\left(a\right)\cos\left(b\right) - \sin\left(a\right)\sin\left(b\right)$

Step 1: Simplify the Expression

To simplify the expression, we need to find the values of $a$ and $b$ . We can do this by using the fact that $\frac{17\pi}{12}$ is equivalent to $\frac{5\pi}{4} + \frac{\pi}{3}$ .

Step 2: Apply the Cosine Sum Identity

Using the cosine sum identity, we can rewrite the expression as:

$\cos\left(\frac{17\pi}{12}\right) = \cos\left(\frac{5\pi}{4}\right)\cos\left(\frac{\pi}{3}\right) - \sin\left(\frac{5\pi}{4}\right)\sin\left(\frac{\pi}{3}\right)$

Step 3: Evaluate the Trigonometric Functions

Now that we have the expression in terms of $a$ and $b$ , we can evaluate the trigonometric functions.

$\cos\left(\frac{5\pi}{4}\right) = -\frac{1}{\sqrt{2}}$
$\sin\left(\frac{5\pi}{4}\right) = -\frac{1}{\sqrt{2}}$
$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$
$\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

Step 4: Substitute the Values

Substituting the values of the trigonometric functions, we get:

$\cos\left(\frac{17\pi}{12}\right) = \left(-\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) - \left(-\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right)$

Step 5: Simplify the Expression

Simplifying the expression, we get:

$\cos\left(\frac{17\pi}{12}\right) = -\frac{1}{2\sqrt{2}} + \frac{\sqrt{3}}{2\sqrt{2}}$

Step 6: Rationalize the Denominator

Rationalizing the denominator, we get:

$\cos\left(\frac{17\pi}{12}\right) = \frac{-1 + \sqrt{3}}{2\sqrt{2}}$

Conclusion

In this article, we used the sine and cosine sum and difference identities to determine the value of the given expression. We simplified the expression by using the fact that $\frac{17\pi}{12}$ is equivalent to $\frac{5\pi}{4} + \frac{\pi}{3}$ . We then evaluated the trigonometric functions and substituted the values into the expression. Finally, we simplified the expression and rationalized the denominator to get the final answer.

Final Answer

The final answer is:

$$\boxed{\frac{-1 + \sqrt{3}}{2\sqrt{2}}}$<br/> **Frequently Asked Questions (FAQs) About Sine and Cosine Sum and Difference Identities**$

Q: What are the sine and cosine sum and difference identities?

A: The sine and cosine sum and difference identities are two fundamental identities in trigonometry that can be used to simplify and solve trigonometric expressions. The sine sum identity is $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ , the sine difference identity is $\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$ , the cosine sum identity is $\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$ , and the cosine difference identity is $\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$ .

Q: How do I use the sine and cosine sum and difference identities to solve trigonometric expressions?

A: To use the sine and cosine sum and difference identities to solve trigonometric expressions, you need to identify the values of $a$ and $b$ in the expression. You can then use the identities to simplify the expression and solve for the unknown values.

Q: What are some common applications of the sine and cosine sum and difference identities?

A: The sine and cosine sum and difference identities have many applications in trigonometry, including solving trigonometric equations, simplifying trigonometric expressions, and finding the values of trigonometric functions.

Q: Can I use the sine and cosine sum and difference identities to solve trigonometric equations?

A: Yes, you can use the sine and cosine sum and difference identities to solve trigonometric equations. By using the identities to simplify the equation, you can solve for the unknown values.

Q: How do I know which identity to use when solving a trigonometric expression?

A: To determine which identity to use when solving a trigonometric expression, you need to identify the values of $a$ and $b$ in the expression. You can then use the identities to simplify the expression and solve for the unknown values.

Q: Can I use the sine and cosine sum and difference identities to find the values of trigonometric functions?

A: Yes, you can use the sine and cosine sum and difference identities to find the values of trigonometric functions. By using the identities to simplify the expression, you can find the values of the trigonometric functions.

Q: What are some common mistakes to avoid when using the sine and cosine sum and difference identities?

A: Some common mistakes to avoid when using the sine and cosine sum and difference identities include:

Not identifying the values of $a$ and $b$ in the expression
Not using the correct identity to simplify the expression
Not simplifying the expression correctly
Not solving for the unknown values

Q: How can I practice using the sine and cosine sum and difference identities?

A: You can practice using the sine and cosine sum and difference identities by working through examples and exercises. You can also use online resources and trigonometry software to help you practice.

Conclusion

In this article, we answered some frequently asked questions about the sine and cosine sum and difference identities. We discussed how to use the identities to solve trigonometric expressions, common applications of the identities, and how to avoid common mistakes. We also provided some tips for practicing using the identities.

Final Answer

The final answer is that the sine and cosine sum and difference identities are powerful tools in trigonometry that can be used to simplify and solve trigonometric expressions. By understanding how to use the identities, you can solve a wide range of trigonometric problems and become proficient in trigonometry.