Evaluate $ \tan \left(-\frac{13 \pi}{4}\right) $.

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Introduction

Trigonometric functions are fundamental in mathematics, and understanding their behavior is crucial for solving various mathematical problems. The tangent function, in particular, is a key component of trigonometry, and evaluating its value for specific angles is essential. In this article, we will focus on evaluating the tangent of a negative angle, specifically $ \tan \left(-\frac{13 \pi}{4}\right) $.

Understanding the Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions:

tan⁡(x)=sin⁡(x)cos⁡(x) \tan(x) = \frac{\sin(x)}{\cos(x)}

This function is periodic with a period of $ \pi $, meaning that the value of the tangent function repeats every $ \pi $ radians. The tangent function is also an odd function, which means that $ \tan(-x) = -\tan(x) $.

Evaluating $ \tan \left(-\frac{13 \pi}{4}\right) $

To evaluate $ \tan \left(-\frac{13 \pi}{4}\right) $, we need to first simplify the angle. We can do this by adding or subtracting multiples of $ \pi $ to the angle until we get an angle between $ -\frac{\pi}{2} $ and $ \frac{\pi}{2} $.

−13π4=−12π4−π4=−3π−π4=−π4 -\frac{13 \pi}{4} = -\frac{12 \pi}{4} - \frac{\pi}{4} = -3 \pi - \frac{\pi}{4} = -\frac{\pi}{4}

Now that we have simplified the angle, we can evaluate the tangent function:

tan⁡(−13π4)=tan⁡(−π4) \tan \left(-\frac{13 \pi}{4}\right) = \tan \left(-\frac{\pi}{4}\right)

Since the tangent function is an odd function, we know that:

tan⁡(−π4)=−tan⁡(π4) \tan \left(-\frac{\pi}{4}\right) = -\tan \left(\frac{\pi}{4}\right)

The value of the tangent function at $ \frac{\pi}{4} $ is 1, so:

tan⁡(−π4)=−1 \tan \left(-\frac{\pi}{4}\right) = -1

Conclusion

In this article, we evaluated the tangent of a negative angle, specifically $ \tan \left(-\frac{13 \pi}{4}\right) $. We simplified the angle by adding or subtracting multiples of $ \pi $ until we got an angle between $ -\frac{\pi}{2} $ and $ \frac{\pi}{2} $. We then used the fact that the tangent function is an odd function to evaluate the tangent function at the simplified angle. The final answer is $ -1 $.

Additional Examples

Evaluating the tangent function for negative angles can be a bit tricky, but with practice, you can become proficient in simplifying the angle and evaluating the tangent function. Here are a few more examples to try:

  • $ \tan \left(-\frac{17 \pi}{4}\right) $
  • $ \tan \left(-\frac{21 \pi}{4}\right) $
  • $ \tan \left(-\frac{25 \pi}{4}\right) $

Try simplifying the angle and evaluating the tangent function for each of these examples.

Common Mistakes

When evaluating the tangent function for negative angles, there are a few common mistakes to watch out for:

  • Not simplifying the angle correctly
  • Not using the fact that the tangent function is an odd function
  • Not evaluating the tangent function at the simplified angle

Make sure to double-check your work and use the correct formulas and properties of the tangent function.

Final Thoughts

Evaluating the tangent function for negative angles can be a bit challenging, but with practice and patience, you can become proficient in simplifying the angle and evaluating the tangent function. Remember to use the fact that the tangent function is an odd function and to evaluate the tangent function at the simplified angle. With these tips and tricks, you'll be able to tackle even the most challenging trigonometry problems.

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by Michael Spivak
  • [3] "Trigonometry for Dummies" by Mary Jane Sterling

Note: The references provided are for general trigonometry and calculus resources. They are not specific to the topic of evaluating the tangent function for negative angles.

Introduction

In our previous article, we evaluated the tangent of a negative angle, specifically $ \tan \left(-\frac{13 \pi}{4}\right) $. We simplified the angle by adding or subtracting multiples of $ \pi $ until we got an angle between $ -\frac{\pi}{2} $ and $ \frac{\pi}{2} $. We then used the fact that the tangent function is an odd function to evaluate the tangent function at the simplified angle. In this article, we will answer some common questions related to evaluating the tangent function for negative angles.

Q: What is the period of the tangent function?

A: The period of the tangent function is $ \pi $, meaning that the value of the tangent function repeats every $ \pi $ radians.

Q: Is the tangent function an odd function?

A: Yes, the tangent function is an odd function, which means that $ \tan(-x) = -\tan(x) $.

Q: How do I simplify a negative angle to get an angle between $ -\frac{\pi}{2} $ and $ \frac{\pi}{2} $?

A: To simplify a negative angle, you can add or subtract multiples of $ \pi $ until you get an angle between $ -\frac{\pi}{2} $ and $ \frac{\pi}{2} $. For example, to simplify $ -\frac{13 \pi}{4} $, you can add $ 4 \pi $ to get $ -\frac{\pi}{4} $.

Q: How do I evaluate the tangent function at a negative angle?

A: To evaluate the tangent function at a negative angle, you can use the fact that the tangent function is an odd function. This means that $ \tan(-x) = -\tan(x) $. For example, to evaluate $ \tan \left(-\frac{\pi}{4}\right) $, you can use the fact that $ \tan \left(-\frac{\pi}{4}\right) = -\tan \left(\frac{\pi}{4}\right) $.

Q: What are some common mistakes to watch out for when evaluating the tangent function for negative angles?

A: Some common mistakes to watch out for when evaluating the tangent function for negative angles include:

  • Not simplifying the angle correctly
  • Not using the fact that the tangent function is an odd function
  • Not evaluating the tangent function at the simplified angle

Q: How can I practice evaluating the tangent function for negative angles?

A: You can practice evaluating the tangent function for negative angles by trying out different examples. Try simplifying the angle and evaluating the tangent function for angles like $ -\frac{17 \pi}{4} $, $ -\frac{21 \pi}{4} $, and $ -\frac{25 \pi}{4} $.

Q: What resources can I use to learn more about evaluating the tangent function for negative angles?

A: There are many resources available to help you learn more about evaluating the tangent function for negative angles. Some recommended resources include:

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak
  • "Trigonometry for Dummies" by Mary Jane Sterling

Conclusion

Evaluating the tangent function for negative angles can be a bit challenging, but with practice and patience, you can become proficient in simplifying the angle and evaluating the tangent function. Remember to use the fact that the tangent function is an odd function and to evaluate the tangent function at the simplified angle. With these tips and tricks, you'll be able to tackle even the most challenging trigonometry problems.

Additional Resources

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by Michael Spivak
  • [3] "Trigonometry for Dummies" by Mary Jane Sterling
  • [4] Khan Academy: Trigonometry
  • [5] MIT OpenCourseWare: Trigonometry

Note: The resources provided are for general trigonometry and calculus resources. They are not specific to the topic of evaluating the tangent function for negative angles.