Solve The Equation: $-\frac{\sqrt{3}}{3} - 2 \tan \theta = -\tan \theta$

**Solve the Equation: $-\frac{\sqrt{3}}{3} - 2 \tan \theta = -\tan \theta$** ===========================================================

Introduction

In this article, we will be solving a trigonometric equation involving the tangent function. The equation is $-\frac{\sqrt{3}}{3} - 2 \tan \theta = -\tan \theta$. We will use various trigonometric identities and properties to simplify and solve the equation.

Understanding the Equation

The given equation is $-\frac{\sqrt{3}}{3} - 2 \tan \theta = -\tan \theta$. We can see that the equation involves the tangent function, which is a fundamental trigonometric function. The tangent function is defined as the ratio of the sine and cosine functions.

Step 1: Isolate the Tangent Function

To solve the equation, we need to isolate the tangent function. We can start by adding $\tan \theta$ to both sides of the equation.

$-\frac{\sqrt{3}}{3} - 2 \tan \theta + \tan \theta = -\tan \theta + \tan \theta$

This simplifies to:

$-\frac{\sqrt{3}}{3} - \tan \theta = 0$

Step 2: Simplify the Equation

Next, we can simplify the equation by adding $\frac{\sqrt{3}}{3}$ to both sides.

$-\frac{\sqrt{3}}{3} + \frac{\sqrt{3}}{3} - \tan \theta = 0 + \frac{\sqrt{3}}{3}$

This simplifies to:

$-\tan \theta = \frac{\sqrt{3}}{3}$

Step 3: Solve for the Tangent Function

To solve for the tangent function, we can multiply both sides of the equation by $-1$.

$\tan \theta = -\frac{\sqrt{3}}{3}$

Step 4: Find the Value of $\theta$

To find the value of $\theta$, we can use the inverse tangent function. The inverse tangent function is denoted by $\tan^{-1}x$.

$\theta = \tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)$

Q&A

Q: What is the tangent function?

A: The tangent function is a fundamental trigonometric function defined as the ratio of the sine and cosine functions.

Q: How do we isolate the tangent function in the equation?

A: We can add $\tan \theta$ to both sides of the equation to isolate the tangent function.

Q: What is the inverse tangent function?

A: The inverse tangent function is denoted by $\tan^{-1}x$ and is used to find the value of $\theta$.

Q: What is the value of $\theta$ in the equation?

A: The value of $\theta$ is $\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)$.

Q: What is the range of the tangent function?

A: The range of the tangent function is all real numbers.

Q: What is the domain of the tangent function?

A: The domain of the tangent function is all real numbers except for odd multiples of $\frac{\pi}{2}$.

Q: How do we simplify the equation?

A: We can add $\frac{\sqrt{3}}{3}$ to both sides of the equation to simplify it.

Q: What is the final answer to the equation?

A: The final answer to the equation is $\tan \theta = -\frac{\sqrt{3}}{3}$.

Conclusion

In this article, we solved the equation $-\frac{\sqrt{3}}{3} - 2 \tan \theta = -\tan \theta$ using various trigonometric identities and properties. We isolated the tangent function, simplified the equation, and found the value of $\theta$. We also answered several questions related to the tangent function and its properties.

Frequently Asked Questions

Q: What is the tangent function?

A: The tangent function is a fundamental trigonometric function defined as the ratio of the sine and cosine functions.

Q: How do we isolate the tangent function in the equation?

A: We can add $\tan \theta$ to both sides of the equation to isolate the tangent function.

Q: What is the inverse tangent function?

A: The inverse tangent function is denoted by $\tan^{-1}x$ and is used to find the value of $\theta$.

Q: What is the value of $\theta$ in the equation?

A: The value of $\theta$ is $\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)$.

Q: What is the range of the tangent function?

A: The range of the tangent function is all real numbers.

Q: What is the domain of the tangent function?

A: The domain of the tangent function is all real numbers except for odd multiples of $\frac{\pi}{2}$.

Q: How do we simplify the equation?

A: We can add $\frac{\sqrt{3}}{3}$ to both sides of the equation to simplify it.

Q: What is the final answer to the equation?

A: The final answer to the equation is $\tan \theta = -\frac{\sqrt{3}}{3}$.