What Is The General Solution To The Trigonometric Equation?$3 \cot \theta = -\sqrt{3}$Drag The Solutions To The Box To Correctly Complete The Table.

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them is crucial for various applications in physics, engineering, and other fields. In this article, we will explore the general solution to the trigonometric equation $3 \cot \theta = -\sqrt{3}$.

Understanding the Equation

The given equation is $3 \cot \theta = -\sqrt{3}$. To solve this equation, we need to isolate the variable $\theta$. We can start by rewriting the equation as $\cot \theta = -\frac{\sqrt{3}}{3}$.

Reciprocal Trigonometric Functions

The cotangent function is the reciprocal of the tangent function. Therefore, we can rewrite the equation as $\tan \theta = -\frac{3}{\sqrt{3}}$.

Simplifying the Equation

We can simplify the equation by rationalizing the denominator. Multiplying both the numerator and denominator by $\sqrt{3}$, we get:

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

Simplifying further, we get:

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

Finding the General Solution

To find the general solution, we need to find the values of $\theta$ that satisfy the equation. We can use the fact that the tangent function has a period of $\pi$ to find the general solution.

The general solution to the equation $\tan \theta = -\sqrt{3}$ is:

$$\theta = \tan^{-1}(-\sqrt{3}) + k\pi$$

where $k$ is an integer.

Using a Calculator or Table

We can use a calculator or table to find the value of $\tan^{-1}(-\sqrt{3})$. The value is approximately $-\frac{\pi}{3}$.

Substituting the Value

Substituting the value of $\tan^{-1}(-\sqrt{3})$ into the general solution, we get:

$$\theta = -\frac{\pi}{3} + k\pi$$

Simplifying the Solution

We can simplify the solution by combining the terms:

$$\theta = k\pi - \frac{\pi}{3}$$

Finding the Solutions

To find the solutions, we need to find the values of $k$ that satisfy the equation. We can do this by substituting different values of $k$ into the solution.

For $k = 0$, we get:

$$\theta = -\frac{\pi}{3}$$

For $k = 1$, we get:

$$\theta = \frac{2\pi}{3}$$

For $k = 2$, we get:

$$\theta = \frac{5\pi}{3}$$

Conclusion

In this article, we have explored the general solution to the trigonometric equation $3 \cot \theta = -\sqrt{3}$. We have used the reciprocal trigonometric functions, simplified the equation, and found the general solution using a calculator or table. We have also found the solutions by substituting different values of $k$ into the solution.

Final Answer

The final answer is:

$$\theta = k\pi - \frac{\pi}{3}$$

where $k$ is an integer.

Drag the Solutions to the Box

Solution	Value
$\theta = -\frac{\pi}{3}$	$-\frac{\pi}{3}$
$\theta = \frac{2\pi}{3}$	$\frac{2\pi}{3}$
$\theta = \frac{5\pi}{3}$	$\frac{5\pi}{3}$

Q: What is the general solution to the trigonometric equation $3 \cot \theta = -\sqrt{3}$?

A: The general solution to the trigonometric equation $3 \cot \theta = -\sqrt{3}$ is $\theta = k\pi - \frac{\pi}{3}$, where $k$ is an integer.

Q: How do I find the solutions to the equation?

A: To find the solutions, you can substitute different values of $k$ into the general solution. For example, for $k = 0$, you get $\theta = -\frac{\pi}{3}$, for $k = 1$, you get $\theta = \frac{2\pi}{3}$, and for $k = 2$, you get $\theta = \frac{5\pi}{3}$.

Q: What is the period of the tangent function?

A: The period of the tangent function is $\pi$. This means that the tangent function repeats itself every $\pi$ radians.

Q: How do I use the reciprocal trigonometric functions to solve the equation?

A: To solve the equation, you can use the reciprocal trigonometric functions. In this case, you can rewrite the equation as $\tan \theta = -\frac{3}{\sqrt{3}}$ and then simplify it to $\tan \theta = -\sqrt{3}$.

Q: What is the value of $\tan^{-1}(-\sqrt{3})$?

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

Q: Can I use a calculator or table to find the value of $\tan^{-1}(-\sqrt{3})$?

A: Yes, you can use a calculator or table to find the value of $\tan^{-1}(-\sqrt{3})$. The value is approximately $-\frac{\pi}{3}$.

Q: How do I simplify the solution?

A: To simplify the solution, you can combine the terms. For example, the solution $\theta = k\pi - \frac{\pi}{3}$ can be simplified to $\theta = k\pi - \frac{\pi}{3}$.

Q: What are the solutions to the equation?

A: The solutions to the equation are $\theta = -\frac{\pi}{3}$, $\theta = \frac{2\pi}{3}$, and $\theta = \frac{5\pi}{3}$.

Q: Can I use the solutions to solve other trigonometric equations?

A: Yes, you can use the solutions to solve other trigonometric equations. The general solution to the trigonometric equation $3 \cot \theta = -\sqrt{3}$ can be used to solve other equations of the form $3 \cot \theta = c$, where $c$ is a constant.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\theta = k\pi - \frac{\pi}{3}$, where $k$ is an integer.

Q: Can I use the final answer to solve other problems?

A: Yes, you can use the final answer to solve other problems. The general solution to the trigonometric equation $3 \cot \theta = -\sqrt{3}$ can be used to solve other problems involving trigonometric equations.

Conclusion

In this article, we have answered some frequently asked questions about the general solution to the trigonometric equation $3 \cot \theta = -\sqrt{3}$. We have covered topics such as the general solution, finding the solutions, the period of the tangent function, using the reciprocal trigonometric functions, and simplifying the solution. We have also provided the final answer to the problem and discussed how to use the final answer to solve other problems.