Find The Exact Value Of The Trigonometric Function. Write Your Answer As A Reduced Fraction.$\[\tan \left(-\frac{5 \pi}{6}\right) =\\]A. \[$\frac{\sqrt{3}}{3}\$\]B. \[$\frac{\sqrt{2}}{3}\$\]C. \[$3 \sqrt{3}\$\]D.

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Introduction

Trigonometric functions are a crucial part of mathematics, and they play a vital role in various fields such as physics, engineering, and navigation. These functions help us describe the relationships between the sides and angles of triangles. In this article, we will focus on finding the exact value of the trigonometric function tan⁑(βˆ’5Ο€6)\tan \left(-\frac{5 \pi}{6}\right).

Understanding the Trigonometric Function

The trigonometric function we are dealing with is the tangent function, denoted by tan⁑\tan. The tangent function is defined as the ratio of the sine and cosine functions. In other words, tan⁑θ=sin⁑θcos⁑θ\tan \theta = \frac{\sin \theta}{\cos \theta}. The tangent function is periodic with a period of Ο€\pi, which means that the value of the tangent function repeats every Ο€\pi radians.

Finding the Exact Value of the Trigonometric Function

To find the exact value of the trigonometric function tan⁑(βˆ’5Ο€6)\tan \left(-\frac{5 \pi}{6}\right), we need to use the properties of the tangent function. We know that the tangent function is an odd function, which means that tan⁑(βˆ’ΞΈ)=βˆ’tan⁑θ\tan (-\theta) = -\tan \theta. Therefore, we can rewrite the given expression as tan⁑(βˆ’5Ο€6)=βˆ’tan⁑(5Ο€6)\tan \left(-\frac{5 \pi}{6}\right) = -\tan \left(\frac{5 \pi}{6}\right).

Using the Unit Circle to Find the Exact Value

To find the exact value of the tangent function, we can use the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions. We can use the unit circle to find the values of the sine and cosine functions, and then use these values to find the value of the tangent function.

Finding the Values of the Sine and Cosine Functions

To find the values of the sine and cosine functions, we need to find the coordinates of the point on the unit circle that corresponds to the angle 5Ο€6\frac{5 \pi}{6}. We can do this by using the properties of the unit circle. The unit circle has a radius of 1, and the coordinates of the point on the unit circle are given by (cos⁑θ,sin⁑θ)(\cos \theta, \sin \theta).

Finding the Value of the Tangent Function

Now that we have found the values of the sine and cosine functions, we can use these values to find the value of the tangent function. The tangent function is defined as the ratio of the sine and cosine functions. Therefore, we can find the value of the tangent function by dividing the value of the sine function by the value of the cosine function.

Calculating the Value of the Tangent Function

To calculate the value of the tangent function, we need to use the values of the sine and cosine functions that we found earlier. We know that sin⁑(5Ο€6)=12\sin \left(\frac{5 \pi}{6}\right) = \frac{1}{2} and cos⁑(5Ο€6)=βˆ’32\cos \left(\frac{5 \pi}{6}\right) = -\frac{\sqrt{3}}{2}. Therefore, we can calculate the value of the tangent function as follows:

tan⁑(5Ο€6)=sin⁑(5Ο€6)cos⁑(5Ο€6)=12βˆ’32=βˆ’13\tan \left(\frac{5 \pi}{6}\right) = \frac{\sin \left(\frac{5 \pi}{6}\right)}{\cos \left(\frac{5 \pi}{6}\right)} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}

Simplifying the Value of the Tangent Function

To simplify the value of the tangent function, we can rationalize the denominator by multiplying the numerator and denominator by 3\sqrt{3}. This gives us:

tan⁑(5Ο€6)=βˆ’13Γ—33=βˆ’33\tan \left(\frac{5 \pi}{6}\right) = -\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{\sqrt{3}}{3}

Finding the Final Answer

Now that we have found the value of the tangent function, we can find the final answer. We know that tan⁑(βˆ’5Ο€6)=βˆ’tan⁑(5Ο€6)\tan \left(-\frac{5 \pi}{6}\right) = -\tan \left(\frac{5 \pi}{6}\right). Therefore, we can find the final answer as follows:

tan⁑(βˆ’5Ο€6)=βˆ’tan⁑(5Ο€6)=βˆ’(βˆ’33)=33\tan \left(-\frac{5 \pi}{6}\right) = -\tan \left(\frac{5 \pi}{6}\right) = -\left(-\frac{\sqrt{3}}{3}\right) = \frac{\sqrt{3}}{3}

Conclusion

In this article, we found the exact value of the trigonometric function tan⁑(βˆ’5Ο€6)\tan \left(-\frac{5 \pi}{6}\right). We used the properties of the tangent function and the unit circle to find the values of the sine and cosine functions, and then used these values to find the value of the tangent function. The final answer is 33\boxed{\frac{\sqrt{3}}{3}}.

Discussion

The tangent function is an important part of mathematics, and it has many applications in various fields such as physics, engineering, and navigation. The unit circle is a powerful tool for finding the values of the trigonometric functions, and it is used extensively in mathematics and science. In this article, we used the unit circle to find the values of the sine and cosine functions, and then used these values to find the value of the tangent function.

Final Answer

The final answer is 33\boxed{\frac{\sqrt{3}}{3}}.

Introduction

In our previous article, we found the exact value of the trigonometric function tan⁑(βˆ’5Ο€6)\tan \left(-\frac{5 \pi}{6}\right). In this article, we will answer some frequently asked questions related to finding the exact value of the trigonometric function.

Q1: What is the tangent function?

A1: The tangent function is a trigonometric function that is defined as the ratio of the sine and cosine functions. It is denoted by tan⁑\tan and is defined as tan⁑θ=sin⁑θcos⁑θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

Q2: How do I find the exact value of the tangent function?

A2: To find the exact value of the tangent function, you need to use the properties of the tangent function and the unit circle. You can use the unit circle to find the values of the sine and cosine functions, and then use these values to find the value of the tangent function.

Q3: What is the unit circle?

A3: The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It is used to define the trigonometric functions and is a powerful tool for finding the values of the sine and cosine functions.

Q4: How do I use the unit circle to find the values of the sine and cosine functions?

A4: To use the unit circle to find the values of the sine and cosine functions, you need to find the coordinates of the point on the unit circle that corresponds to the angle you are interested in. The coordinates of the point on the unit circle are given by (cos⁑θ,sin⁑θ)(\cos \theta, \sin \theta).

Q5: What is the relationship between the sine and cosine functions?

A5: The sine and cosine functions are related by the Pythagorean identity, which states that sin⁑2θ+cos⁑2θ=1\sin^2 \theta + \cos^2 \theta = 1. This identity can be used to find the values of the sine and cosine functions.

Q6: How do I find the value of the tangent function using the unit circle?

A6: To find the value of the tangent function using the unit circle, you need to find the values of the sine and cosine functions, and then use these values to find the value of the tangent function. The tangent function is defined as the ratio of the sine and cosine functions, so you can find the value of the tangent function by dividing the value of the sine function by the value of the cosine function.

Q7: What is the final answer to the problem?

A7: The final answer to the problem is 33\boxed{\frac{\sqrt{3}}{3}}.

Q8: Can I use the tangent function to find the values of other trigonometric functions?

A8: Yes, you can use the tangent function to find the values of other trigonometric functions. The tangent function is related to the sine and cosine functions, so you can use the tangent function to find the values of the sine and cosine functions.

Q9: How do I apply the tangent function in real-world problems?

A9: The tangent function has many applications in real-world problems, such as physics, engineering, and navigation. You can use the tangent function to find the values of the sine and cosine functions, and then use these values to solve problems in these fields.

Q10: What are some common mistakes to avoid when finding the exact value of the tangent function?

A10: Some common mistakes to avoid when finding the exact value of the tangent function include:

  • Not using the unit circle to find the values of the sine and cosine functions
  • Not using the Pythagorean identity to find the values of the sine and cosine functions
  • Not simplifying the expression for the tangent function
  • Not checking the final answer for errors

Conclusion

In this article, we answered some frequently asked questions related to finding the exact value of the trigonometric function. We hope that this article has been helpful in clarifying any confusion you may have had about finding the exact value of the tangent function. If you have any further questions, please don't hesitate to ask.