Express Tan ⁡ 310 ∘ \tan 310^{\circ} Tan 31 0 ∘ As A Function Of A Positive Acute Angle.

Introduction

Trigonometric functions are essential in mathematics, and understanding how to express them in terms of acute angles is crucial for solving various problems. In this article, we will focus on expressing the tangent function in terms of a positive acute angle. We will start by understanding the properties of the tangent function and then use these properties to express $\tan 310^{\circ}$ as a function of a positive acute angle.

Understanding the Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions:

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

The tangent function has a periodicity of $180^{\circ}$, which means that the value of the tangent function repeats every $180^{\circ}$. This property is essential in expressing the tangent function in terms of acute angles.

Expressing $\tan 310^{\circ}$ as a Function of a Positive Acute Angle

To express $\tan 310^{\circ}$ as a function of a positive acute angle, we can use the periodicity property of the tangent function. We know that the tangent function has a period of $180^{\circ}$, so we can subtract multiples of $180^{\circ}$ from the given angle to obtain an equivalent angle within the range of $0^{\circ}$ to $180^{\circ}$.

Let's start by subtracting $360^{\circ}$ from $310^{\circ}$:

$$310^{\circ} - 360^{\circ} = -50^{\circ}$$

Since the tangent function is an odd function, we can express $\tan (-50^{\circ})$ as $-\tan 50^{\circ}$:

$$\tan (-50^{\circ}) = -\tan 50^{\circ}$$

Now, we can use the fact that the tangent function is an odd function to express $\tan 310^{\circ}$ as a function of a positive acute angle:

$$\tan 310^{\circ} = \tan (-50^{\circ}) = -\tan 50^{\circ}$$

Conclusion

In this article, we have shown how to express the tangent function in terms of a positive acute angle. We have used the periodicity property of the tangent function to express $\tan 310^{\circ}$ as a function of a positive acute angle. The result is $\tan 310^{\circ} = -\tan 50^{\circ}$.

Properties of the Tangent Function

The tangent function has several important properties that are essential in expressing it in terms of acute angles. Some of these properties include:

• Periodicity: The tangent function has a periodicity of $180^{\circ}$, which means that the value of the tangent function repeats every $180^{\circ}$.
• Odd Function: The tangent function is an odd function, which means that $\tan (-\theta) = -\tan \theta$.
• Range: The range of the tangent function is all real numbers.

Applications of the Tangent Function

The tangent function has several important applications in mathematics and physics. Some of these applications include:

• Trigonometric Identities: The tangent function is used to derive various trigonometric identities, such as the Pythagorean identity.
• Circular Functions: The tangent function is used to define circular functions, such as the sine and cosine functions.
• Physics: The tangent function is used to describe the motion of objects in physics, such as the motion of a projectile.

Solving Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions. Solving trigonometric equations involves using various techniques, such as the tangent function, to find the solutions to the equation.

Example 1

Solve the equation $\tan \theta = 2$.

To solve this equation, we can use the fact that the tangent function is an odd function:

$$\tan \theta = 2 \Rightarrow \tan (-\theta) = -2$$

Now, we can use the periodicity property of the tangent function to find the solutions to the equation:

$$\theta = \tan^{-1} 2 \Rightarrow -\theta = \tan^{-1} (-2)$$

The solutions to the equation are $\theta = \tan^{-1} 2$ and $-\theta = \tan^{-1} (-2)$.

Example 2

Solve the equation $\tan \theta = -2$.

To solve this equation, we can use the fact that the tangent function is an odd function:

$$\tan \theta = -2 \Rightarrow \tan (-\theta) = 2$$

Now, we can use the periodicity property of the tangent function to find the solutions to the equation:

$$\theta = \tan^{-1} (-2) \Rightarrow -\theta = \tan^{-1} 2$$

The solutions to the equation are $\theta = \tan^{-1} (-2)$ and $-\theta = \tan^{-1} 2$.

Conclusion

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Q: What is the periodicity of the tangent function?

A: The tangent function has a periodicity of $180^{\circ}$, which means that the value of the tangent function repeats every $180^{\circ}$.

Q: How do I express $\tan 310^{\circ}$ as a function of a positive acute angle?

A: To express $\tan 310^{\circ}$ as a function of a positive acute angle, you can use the periodicity property of the tangent function. Subtract multiples of $180^{\circ}$ from the given angle to obtain an equivalent angle within the range of $0^{\circ}$ to $180^{\circ}$. In this case, we can subtract $360^{\circ}$ from $310^{\circ}$ to get $-50^{\circ}$. Since the tangent function is an odd function, we can express $\tan (-50^{\circ})$ as $-\tan 50^{\circ}$.

Q: What is the range of the tangent function?

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

Q: How do I solve trigonometric equations involving the tangent function?

A: To solve trigonometric equations involving the tangent function, you can use various techniques, such as the tangent function, to find the solutions to the equation. For example, if you have the equation $\tan \theta = 2$, you can use the fact that the tangent function is an odd function to express $\tan (-\theta) = -2$. Then, you can use the periodicity property of the tangent function to find the solutions to the equation.

Q: What are some common applications of the tangent function?

A: The tangent function has several important applications in mathematics and physics, such as:

• Trigonometric Identities: The tangent function is used to derive various trigonometric identities, such as the Pythagorean identity.
• Circular Functions: The tangent function is used to define circular functions, such as the sine and cosine functions.
• Physics: The tangent function is used to describe the motion of objects in physics, such as the motion of a projectile.

Q: How do I use the tangent function to solve problems involving right triangles?

A: To use the tangent function to solve problems involving right triangles, you can use the fact that the tangent function is equal to the ratio of the opposite side to the adjacent side. For example, if you have a right triangle with a hypotenuse of length 10 and an opposite side of length 6, you can use the tangent function to find the length of the adjacent side.

Q: What are some common mistakes to avoid when working with the tangent function?

A: Some common mistakes to avoid when working with the tangent function include:

• Not using the periodicity property of the tangent function: Make sure to use the periodicity property of the tangent function to express the tangent function in terms of acute angles.
• Not using the fact that the tangent function is an odd function: Make sure to use the fact that the tangent function is an odd function to express the tangent function in terms of acute angles.
• Not checking the range of the tangent function: Make sure to check the range of the tangent function to ensure that the solution is valid.

Q: How do I graph the tangent function?

A: To graph the tangent function, you can use a graphing calculator or a computer program to plot the function. You can also use the fact that the tangent function is equal to the ratio of the opposite side to the adjacent side to graph the function.

Q: What are some common applications of the tangent function in real-world problems?

A: The tangent function has several important applications in real-world problems, such as:

• Navigation: The tangent function is used in navigation to calculate the direction of a ship or an airplane.
• Surveying: The tangent function is used in surveying to calculate the distance between two points.
• Physics: The tangent function is used in physics to describe the motion of objects, such as the motion of a projectile.