Find All Angles, $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$, That Solve The Following Equation:$\tan \theta = -\frac{\sqrt{3}}{3}$

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Introduction

In trigonometry, the tangent function is a fundamental concept used to relate the angles of a right-angled triangle to the ratios of the lengths of its sides. The tangent function 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 this article, we will explore the angles that satisfy the equation $\tan \theta = -\frac{\sqrt{3}}{3}$, where $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$.

Understanding the Tangent Function

The tangent function is a periodic function, meaning that it repeats its values at regular intervals. The period of the tangent function is $180^{\circ}$, which means that the function repeats its values every $180^{\circ}$. This is because the tangent function is defined as the ratio of the sine and cosine functions, which are periodic with a period of $360^{\circ}$.

Solving the Equation

To solve the equation $\tan \theta = -\frac{\sqrt{3}}{3}$, we need to find the angles that satisfy this equation. We can start by using the inverse tangent function, denoted as $\tan^{-1}$, to find the angle that satisfies the equation. However, we need to be careful when using the inverse tangent function, as it is only defined for angles in the range $(-90^{\circ}, 90^{\circ})$.

Using the Inverse Tangent Function

Using the inverse tangent function, we can find the angle that satisfies the equation $\tan \theta = -\frac{\sqrt{3}}{3}$. We have:

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

However, we need to be careful when using the inverse tangent function, as it is only defined for angles in the range $(-90^{\circ}, 90^{\circ})$. To find the angles that satisfy the equation, we need to use the properties of the tangent function.

Properties of the Tangent Function

The tangent function has several important properties that we can use to find the angles that satisfy the equation. One of the most important properties is that the tangent function is an odd function, meaning that $\tan (-\theta) = -\tan \theta$. This means that if $\theta$ is an angle that satisfies the equation, then $-\theta$ is also an angle that satisfies the equation.

Finding the Angles

Using the properties of the tangent function, we can find the angles that satisfy the equation $\tan \theta = -\frac{\sqrt{3}}{3}$. We have:

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

Using the property that the tangent function is an odd function, we can write:

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

This means that $-\theta$ is an angle that satisfies the equation $\tan \theta = \frac{\sqrt{3}}{3}$. We know that the angle that satisfies this equation is $60^{\circ}$, so we have:

$$-\theta = 60^{\circ}$$

Solving for $\theta$, we get:

$$\theta = -60^{\circ}$$

However, we need to find the angles that satisfy the equation in the range $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$. To do this, we can add $180^{\circ}$ to the angle, which gives us:

$$\theta = -60^{\circ} + 180^{\circ} = 120^{\circ}$$

This means that the angle that satisfies the equation is $120^{\circ}$.

Finding the Second Angle

We have found one angle that satisfies the equation, but we need to find the second angle. To do this, we can use the property that the tangent function is periodic with a period of $180^{\circ}$. This means that if $\theta$ is an angle that satisfies the equation, then $\theta + 180^{\circ}$ is also an angle that satisfies the equation.

Using the Periodic Property

Using the periodic property, we can write:

$$\tan (\theta + 180^{\circ}) = \tan \theta = -\frac{\sqrt{3}}{3}$$

This means that $\theta + 180^{\circ}$ is an angle that satisfies the equation. We know that the angle that satisfies this equation is $120^{\circ}$, so we have:

$$\theta + 180^{\circ} = 120^{\circ}$$

Solving for $\theta$, we get:

$$\theta = -60^{\circ}$$

However, we need to find the second angle that satisfies the equation in the range $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$. To do this, we can add $360^{\circ}$ to the angle, which gives us:

$$\theta = -60^{\circ} + 360^{\circ} = 300^{\circ}$$

This means that the second angle that satisfies the equation is $300^{\circ}$.

Conclusion

In this article, we have found the angles that satisfy the equation $\tan \theta = -\frac{\sqrt{3}}{3}$, where $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$. We have used the properties of the tangent function, including the fact that it is an odd function and periodic with a period of $180^{\circ}$. We have found two angles that satisfy the equation, which are $120^{\circ}$ and $300^{\circ}$.

Final Answer

The final answer is $\boxed{120^{\circ}, 300^{\circ}}$.

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Introduction

In our previous article, we explored the angles that satisfy the equation $\tan \theta = -\frac{\sqrt{3}}{3}$, where $0^{\circ} \leq \theta \ \textless \ 360^{\circ}$. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the tangent function?

A: The tangent function is a fundamental concept in trigonometry that relates the angles of a right-angled triangle to the ratios of the lengths of its sides. 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.

Q: What is the period of the tangent function?

A: The period of the tangent function is $180^{\circ}$, meaning that the function repeats its values every $180^{\circ}$.

Q: How do I find the angles that satisfy the equation $\tan \theta = -\frac{\sqrt{3}}{3}$?

A: To find the angles that satisfy the equation, you can use the properties of the tangent function, including the fact that it is an odd function and periodic with a period of $180^{\circ}$. You can also use the inverse tangent function, denoted as $\tan^{-1}$, to find the angle that satisfies the equation.

Q: What are the two angles that satisfy the equation $\tan \theta = -\frac{\sqrt{3}}{3}$?

A: The two angles that satisfy the equation are $120^{\circ}$ and $300^{\circ}$.

Q: Why do we need to add $180^{\circ}$ to the angle to find the second angle?

A: We need to add $180^{\circ}$ to the angle to find the second angle because the tangent function is periodic with a period of $180^{\circ}$. This means that if $\theta$ is an angle that satisfies the equation, then $\theta + 180^{\circ}$ is also an angle that satisfies the equation.

Q: Can I use the inverse tangent function to find the angles that satisfy the equation?

A: Yes, you can use the inverse tangent function to find the angles that satisfy the equation. However, you need to be careful when using the inverse tangent function, as it is only defined for angles in the range $(-90^{\circ}, 90^{\circ})$.

Q: What is the significance of the angle $60^{\circ}$ in this problem?

A: The angle $60^{\circ}$ is significant in this problem because it is the angle that satisfies the equation $\tan \theta = \frac{\sqrt{3}}{3}$. We can use this angle to find the second angle that satisfies the equation by adding $180^{\circ}$ to it.

Q: Can I use the tangent function to find the angles that satisfy the equation in other ranges?

A: Yes, you can use the tangent function to find the angles that satisfy the equation in other ranges. However, you need to be careful when using the tangent function, as it is periodic with a period of $180^{\circ}$.

Conclusion

In this article, we have answered some of the most frequently asked questions related to the angles that satisfy the equation $\tan \theta = -\frac{\sqrt{3}}{3}$. We hope that this article has been helpful in clarifying any doubts that you may have had.

Final Answer

The final answer is $\boxed{120^{\circ}, 300^{\circ}}$.