Find The Exact Value Of $\tan \frac{-\pi}{6}$.

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental concepts in trigonometry is the tangent function, which is defined as the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle. In this article, we will explore how to find the exact value of $\tan \frac{-\pi}{6}$.

Understanding the Tangent Function

The tangent function is a periodic function, which means that it repeats itself at regular intervals. The period of the tangent function is $\pi$, which means that the function repeats itself every $\pi$ radians. The tangent function can be defined as:

$$\tan x = \frac{\sin x}{\cos x}$$

where $x$ is the angle in radians.

Finding the Exact Value of $\tan \frac{-\pi}{6}$

To find the exact value of $\tan \frac{-\pi}{6}$, we need to use the definition of the tangent function and the properties of the sine and cosine functions.

We know that the sine and cosine functions are periodic functions with a period of $2\pi$. This means that the values of the sine and cosine functions repeat themselves every $2\pi$ radians.

Using the definition of the tangent function, we can write:

$$\tan \frac{-\pi}{6} = \frac{\sin \frac{-\pi}{6}}{\cos \frac{-\pi}{6}}$$

To find the exact value of $\tan \frac{-\pi}{6}$, we need to find the exact values of $\sin \frac{-\pi}{6}$ and $\cos \frac{-\pi}{6}$.

Finding the Exact Value of $\sin \frac{-\pi}{6}$

To find the exact value of $\sin \frac{-\pi}{6}$, we can use the fact that the sine function is an odd function. This means that:

$$\sin (-x) = -\sin x$$

Using this property, we can write:

$$\sin \frac{-\pi}{6} = -\sin \frac{\pi}{6}$$

We know that the value of $\sin \frac{\pi}{6}$ is $\frac{1}{2}$. Therefore, we can write:

$$\sin \frac{-\pi}{6} = -\frac{1}{2}$$

Finding the Exact Value of $\cos \frac{-\pi}{6}$

To find the exact value of $\cos \frac{-\pi}{6}$, we can use the fact that the cosine function is an even function. This means that:

$$\cos (-x) = \cos x$$

Using this property, we can write:

$$\cos \frac{-\pi}{6} = \cos \frac{\pi}{6}$$

We know that the value of $\cos \frac{\pi}{6}$ is $\frac{\sqrt{3}}{2}$. Therefore, we can write:

$$\cos \frac{-\pi}{6} = \frac{\sqrt{3}}{2}$$

Finding the Exact Value of $\tan \frac{-\pi}{6}$

Now that we have found the exact values of $\sin \frac{-\pi}{6}$ and $\cos \frac{-\pi}{6}$, we can find the exact value of $\tan \frac{-\pi}{6}$.

Using the definition of the tangent function, we can write:

$$\tan \frac{-\pi}{6} = \frac{\sin \frac{-\pi}{6}}{\cos \frac{-\pi}{6}}$$

Substituting the values of $\sin \frac{-\pi}{6}$ and $\cos \frac{-\pi}{6}$, we get:

$$\tan \frac{-\pi}{6} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}$$

Simplifying the expression, we get:

$$\tan \frac{-\pi}{6} = -\frac{1}{\sqrt{3}}$$

Conclusion

In this article, we have explored how to find the exact value of $\tan \frac{-\pi}{6}$. We have used the definition of the tangent function and the properties of the sine and cosine functions to find the exact values of $\sin \frac{-\pi}{6}$ and $\cos \frac{-\pi}{6}$. We have then used these values to find the exact value of $\tan \frac{-\pi}{6}$. The exact value of $\tan \frac{-\pi}{6}$ is $-\frac{1}{\sqrt{3}}$.

References

• "Trigonometry" by Michael Corral
• "Calculus" by Michael Spivak
• "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• "Trigonometry: A Unit Circle Approach" by Michael Sullivan
• "Calculus: Early Transcendentals" by James Stewart
• "Mathematics: A Very Short Introduction" by Timothy Gowers
  

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Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle. It can be written as:

$$\tan x = \frac{\sin x}{\cos x}$$

Q: Why is the tangent function important?

A: The tangent function is important in trigonometry because it is used to solve problems involving right triangles. It is also used in calculus to find the derivative of the sine and cosine functions.

Q: How do you find the exact value of $\tan \frac{-\pi}{6}$?

A: To find the exact value of $\tan \frac{-\pi}{6}$, you need to use the definition of the tangent function and the properties of the sine and cosine functions. You can start by finding the exact values of $\sin \frac{-\pi}{6}$ and $\cos \frac{-\pi}{6}$, and then use these values to find the exact value of $\tan \frac{-\pi}{6}$.

Q: What are the exact values of $\sin \frac{-\pi}{6}$ and $\cos \frac{-\pi}{6}$?

A: The exact value of $\sin \frac{-\pi}{6}$ is $-\frac{1}{2}$, and the exact value of $\cos \frac{-\pi}{6}$ is $\frac{\sqrt{3}}{2}$.

Q: How do you simplify the expression $\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}$?

A: To simplify the expression $\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}$, you can multiply the numerator and denominator by 2 to get:

$$\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{-1}{\sqrt{3}}$$

Q: What is the final answer for the exact value of $\tan \frac{-\pi}{6}$?

A: The final answer for the exact value of $\tan \frac{-\pi}{6}$ is $-\frac{1}{\sqrt{3}}$.

Q: Can you provide more examples of finding the exact value of the tangent function?

A: Yes, here are a few more examples:

• $\tan \frac{\pi}{4} = 1$
• $\tan \frac{\pi}{3} = \sqrt{3}$
• $\tan \frac{\pi}{2} = \infty$

Q: How do you use the tangent function in real-world applications?

A: The tangent function is used in a variety of real-world applications, including:

• Navigation: The tangent function is used to calculate the angle of elevation or depression of an object.
• Physics: The tangent function is used to calculate the angle of incidence or reflection of a wave.
• Engineering: The tangent function is used to calculate the angle of a slope or a beam.

Q: Can you provide more resources for learning about the tangent function?

A: Yes, here are a few more resources:

• "Trigonometry" by Michael Corral
• "Calculus" by Michael Spivak
• "Mathematics for the Nonmathematician" by Morris Kline

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

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

• Not using the correct definition of the tangent function.
• Not simplifying the expression correctly.
• Not using the properties of the sine and cosine functions correctly.

Q: How do you check your work when finding the exact value of the tangent function?

A: To check your work when finding the exact value of the tangent function, you can use the following steps:

• Plug in the values of the sine and cosine functions into the definition of the tangent function.
• Simplify the expression using the properties of the sine and cosine functions.
• Check that the final answer is in the correct form.