Use An Addition Or Subtraction Formula To Find The Exact Value Of The Expression.$\tan \left(\frac{17 \pi}{12}\right$\]

**Use an Addition or Subtraction Formula to Find the Exact Value of the Expression** ===========================================================

Introduction

In trigonometry, addition and subtraction formulas are used to simplify and evaluate trigonometric expressions. These formulas allow us to express a trigonometric function of a sum or difference of two angles in terms of the trigonometric functions of the individual angles. In this article, we will use the addition and subtraction formulas to find the exact value of the expression $\tan \left(\frac{17 \pi}{12}\right)$.

What are Addition and Subtraction Formulas?

Addition and subtraction formulas are used to express a trigonometric function of a sum or difference of two angles in terms of the trigonometric functions of the individual angles. These formulas are:

• $\sin (A + B) = \sin A \cos B + \cos A \sin B$
• $\sin (A - B) = \sin A \cos B - \cos A \sin B$
• $\cos (A + B) = \cos A \cos B - \sin A \sin B$
• $\cos (A - B) = \cos A \cos B + \sin A \sin B$
• $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
• $\tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

How to Use Addition and Subtraction Formulas

To use the addition and subtraction formulas, we need to identify the angles A and B in the given expression. We can then substitute these values into the formulas and simplify the expression.

Example: Finding the Exact Value of $\tan \left(\frac{17 \pi}{12}\right)$

To find the exact value of $\tan \left(\frac{17 \pi}{12}\right)$, we can use the addition formula for tangent:

$\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

We can rewrite $\frac{17 \pi}{12}$ as $\frac{4 \pi}{3} + \frac{2 \pi}{3}$.

$\tan \left(\frac{17 \pi}{12}\right) = \tan \left(\frac{4 \pi}{3} + \frac{2 \pi}{3}\right)$

Using the addition formula for tangent, we get:

$\tan \left(\frac{17 \pi}{12}\right) = \frac{\tan \left(\frac{4 \pi}{3}\right) + \tan \left(\frac{2 \pi}{3}\right)}{1 - \tan \left(\frac{4 \pi}{3}\right) \tan \left(\frac{2 \pi}{3}\right)}$

We know that $\tan \left(\frac{4 \pi}{3}\right) = -\sqrt{3}$ and $\tan \left(\frac{2 \pi}{3}\right) = \sqrt{3}$.

Substituting these values into the formula, we get:

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

Simplifying the expression, we get:

$\tan \left(\frac{17 \pi}{12}\right) = \frac{0}{1 + 3}$

$\tan \left(\frac{17 \pi}{12}\right) = \frac{0}{4}$

$\tan \left(\frac{17 \pi}{12}\right) = 0$

Conclusion

In this article, we used the addition and subtraction formulas to find the exact value of the expression $\tan \left(\frac{17 \pi}{12}\right)$. We rewrote the expression as a sum of two angles and used the addition formula for tangent to simplify the expression. We then substituted the values of the tangent functions into the formula and simplified the expression to get the final answer.

Frequently Asked Questions

Q: What are addition and subtraction formulas?

A: Addition and subtraction formulas are used to express a trigonometric function of a sum or difference of two angles in terms of the trigonometric functions of the individual angles.

Q: How do I use addition and subtraction formulas?

A: To use the addition and subtraction formulas, you need to identify the angles A and B in the given expression. You can then substitute these values into the formulas and simplify the expression.

Q: What is the exact value of $\tan \left(\frac{17 \pi}{12}\right)$?

A: The exact value of $\tan \left(\frac{17 \pi}{12}\right)$ is 0.

Q: How do I simplify a trigonometric expression using addition and subtraction formulas?

A: To simplify a trigonometric expression using addition and subtraction formulas, you need to identify the angles A and B in the given expression. You can then substitute these values into the formulas and simplify the expression.

Q: What are the addition and subtraction formulas for sine, cosine, and tangent?

A: The addition and subtraction formulas for sine, cosine, and tangent are:

• $\sin (A + B) = \sin A \cos B + \cos A \sin B$
• $\sin (A - B) = \sin A \cos B - \cos A \sin B$
• $\cos (A + B) = \cos A \cos B - \sin A \sin B$
• $\cos (A - B) = \cos A \cos B + \sin A \sin B$
• $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
• $\tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$