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Introduction

In trigonometry, equations involving the tangent function are common and can be used to find the angles between two lines or in various geometric problems. In this article, we will focus on finding all angles $\theta$ between $0^{\circ}$ and $180^{\circ}$ satisfying the given equation. We will use the tangent function and its properties to solve the equation and find the required angles.

The Given Equation

The given equation is $\tan \theta = \frac{1}{3}$. This equation involves the tangent function, which is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.

Understanding the Tangent Function

The tangent function is a periodic function with a period of $180^{\circ}$. This means that the tangent function repeats its values every $180^{\circ}$. The range of the tangent function is all real numbers, and its domain is all real numbers except for odd multiples of $90^{\circ}$.

Solving the Equation

To solve the equation $\tan \theta = \frac{1}{3}$, we need to find the angles $\theta$ that satisfy this equation. We can use the inverse tangent function, also known as the arctangent function, to find the angles. The arctangent function is defined as the inverse of the tangent function and is used to find the angle whose tangent is a given value.

Finding the Angles

Using the arctangent function, we can find the angles $\theta$ that satisfy the equation $\tan \theta = \frac{1}{3}$. The arctangent function is defined as:

$$\theta = \arctan \left(\frac{1}{3}\right)$$

Evaluating this expression, we get:

$$\theta \approx 18.43^{\circ}$$

However, since the tangent function is periodic with a period of $180^{\circ}$, we need to find all angles between $0^{\circ}$ and $180^{\circ}$ that satisfy the equation. We can do this by adding multiples of $180^{\circ}$ to the angle we found.

Finding All Angles

To find all angles between $0^{\circ}$ and $180^{\circ}$ that satisfy the equation, we can add multiples of $180^{\circ}$ to the angle we found. This gives us:

$$\theta \approx 18.43^{\circ}, 198.57^{\circ}$$

However, we need to round our answer to one decimal place. Rounding these values, we get:

$$\theta \approx 18.4^{\circ}, 198.6^{\circ}$$

Conclusion

In this article, we found all angles $\theta$ between $0^{\circ}$ and $180^{\circ}$ satisfying the given equation. We used the tangent function and its properties to solve the equation and find the required angles. We also used the arctangent function to find the angles and added multiples of $180^{\circ}$ to find all angles between $0^{\circ}$ and $180^{\circ}$ that satisfy the equation.

Final Answer

The final answer is $\boxed{18.4^{\circ}, 198.6^{\circ}}$.

Introduction

In our previous article, we discussed finding all angles $\theta$ between $0^{\circ}$ and $180^{\circ}$ satisfying the given equation $\tan \theta = \frac{1}{3}$. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the tangent function?

A: The tangent function is a periodic function with a period of $180^{\circ}$. It is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.

Q: How do I use the arctangent function to find the angles?

A: To use the arctangent function, you need to find the angle whose tangent is a given value. The arctangent function is defined as:

$$\theta = \arctan \left(\frac{1}{3}\right)$$

Evaluating this expression, you get:

$$\theta \approx 18.43^{\circ}$$

Q: Why do I need to add multiples of $180^{\circ}$ to the angle?

A: Since the tangent function is periodic with a period of $180^{\circ}$, you need to add multiples of $180^{\circ}$ to the angle to find all angles between $0^{\circ}$ and $180^{\circ}$ that satisfy the equation.

Q: How do I round my answer to one decimal place?

A: To round your answer to one decimal place, you need to look at the second decimal place. If it is less than 5, you round down. If it is 5 or greater, you round up.

Q: What if I have a different equation, such as $\tan \theta = \frac{2}{3}$?

A: To solve a different equation, you need to use the arctangent function and add multiples of $180^{\circ}$ to the angle. For example, to solve the equation $\tan \theta = \frac{2}{3}$, you would use:

$$\theta = \arctan \left(\frac{2}{3}\right)$$

Evaluating this expression, you get:

$$\theta \approx 33.69^{\circ}$$

Adding multiples of $180^{\circ}$, you get:

$$\theta \approx 33.7^{\circ}, 153.3^{\circ}$$

Q: Can I use a calculator to find the angles?

A: Yes, you can use a calculator to find the angles. Most calculators have a tangent function and an arctangent function that you can use to find the angles.

Conclusion

In this article, we answered some frequently asked questions related to finding angles satisfying the given equation. We discussed the tangent function, the arctangent function, and how to use them to find the angles. We also answered questions about rounding the answer to one decimal place and using a calculator to find the angles.

Final Answer

The final answer is $\boxed{18.4^{\circ}, 198.6^{\circ}}$.