$\operatorname{arccot}\left(\frac{1}{3} \sqrt{3}\right) =$

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Introduction

The inverse cotangent function, denoted as arccot⁑(x)\operatorname{arccot}(x), is the inverse of the cotangent function. It is a mathematical function that returns the angle whose cotangent is a given value. In this article, we will explore the solution to the equation arccot⁑(133)=\operatorname{arccot}\left(\frac{1}{3} \sqrt{3}\right) =. We will break down the problem into smaller steps and provide a clear explanation of each step.

Understanding the Cotangent Function

Before we dive into the solution, let's briefly review the cotangent function. The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle. It is denoted as cot⁑(x)\cot(x) and is equal to cos⁑(x)sin⁑(x)\frac{\cos(x)}{\sin(x)}. The cotangent function has a period of Ο€\pi, which means that it repeats every Ο€\pi radians.

The Inverse Cotangent Function

The inverse cotangent function, denoted as arccot⁑(x)\operatorname{arccot}(x), is the inverse of the cotangent function. It returns the angle whose cotangent is a given value. The range of the inverse cotangent function is (βˆ’Ο€2,Ο€2)(-\frac{\pi}{2}, \frac{\pi}{2}), which means that it returns an angle between βˆ’Ο€2-\frac{\pi}{2} and Ο€2\frac{\pi}{2} radians.

Solving the Equation

Now that we have a good understanding of the cotangent and inverse cotangent functions, let's solve the equation arccot⁑(133)=\operatorname{arccot}\left(\frac{1}{3} \sqrt{3}\right) =. To solve this equation, we need to find the angle whose cotangent is equal to 133\frac{1}{3} \sqrt{3}.

Step 1: Simplify the Expression

The first step is to simplify the expression 133\frac{1}{3} \sqrt{3}. We can rewrite this expression as 33\frac{\sqrt{3}}{3}.

Step 2: Find the Cotangent of the Angle

The next step is to find the cotangent of the angle whose cotangent is equal to 33\frac{\sqrt{3}}{3}. We can use the definition of the cotangent function to find this angle.

cot⁑(x)=cos⁑(x)sin⁑(x)\cot(x) = \frac{\cos(x)}{\sin(x)}

We know that cot⁑(x)=33\cot(x) = \frac{\sqrt{3}}{3}, so we can set up the following equation:

cos⁑(x)sin⁑(x)=33\frac{\cos(x)}{\sin(x)} = \frac{\sqrt{3}}{3}

Step 3: Solve for the Angle

To solve for the angle, we can use the following trigonometric identity:

cos⁑(x)=11+tan⁑2(x)\cos(x) = \frac{1}{\sqrt{1 + \tan^2(x)}}

We can substitute this expression into the equation above:

11+tan⁑2(x)sin⁑(x)=33\frac{\frac{1}{\sqrt{1 + \tan^2(x)}}}{\sin(x)} = \frac{\sqrt{3}}{3}

Step 4: Simplify the Equation

We can simplify the equation above by multiplying both sides by sin⁑(x)\sin(x):

11+tan⁑2(x)=33sin⁑(x)\frac{1}{\sqrt{1 + \tan^2(x)}} = \frac{\sqrt{3}}{3} \sin(x)

Step 5: Solve for the Angle

To solve for the angle, we can square both sides of the equation:

11+tan⁑2(x)=39sin⁑2(x)\frac{1}{1 + \tan^2(x)} = \frac{3}{9} \sin^2(x)

We can simplify the right-hand side of the equation:

11+tan⁑2(x)=13sin⁑2(x)\frac{1}{1 + \tan^2(x)} = \frac{1}{3} \sin^2(x)

Step 6: Solve for the Angle

To solve for the angle, we can use the following trigonometric identity:

tan⁑2(x)=sin⁑2(x)cos⁑2(x)\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}

We can substitute this expression into the equation above:

11+sin⁑2(x)cos⁑2(x)=13sin⁑2(x)\frac{1}{1 + \frac{\sin^2(x)}{\cos^2(x)}} = \frac{1}{3} \sin^2(x)

Step 7: Simplify the Equation

We can simplify the equation above by multiplying both sides by 1+sin⁑2(x)cos⁑2(x)1 + \frac{\sin^2(x)}{\cos^2(x)}:

1=13sin⁑2(x)+sin⁑2(x)cos⁑2(x)1 = \frac{1}{3} \sin^2(x) + \frac{\sin^2(x)}{\cos^2(x)}

Step 8: Solve for the Angle

To solve for the angle, we can use the following trigonometric identity:

sin⁑2(x)+cos⁑2(x)=1\sin^2(x) + \cos^2(x) = 1

We can substitute this expression into the equation above:

1=13(sin⁑2(x)+cos⁑2(x))+sin⁑2(x)cos⁑2(x)1 = \frac{1}{3} (\sin^2(x) + \cos^2(x)) + \frac{\sin^2(x)}{\cos^2(x)}

Step 9: Simplify the Equation

We can simplify the equation above by multiplying both sides by 33:

3=sin⁑2(x)+cos⁑2(x)+3sin⁑2(x)cos⁑2(x)3 = \sin^2(x) + \cos^2(x) + 3 \frac{\sin^2(x)}{\cos^2(x)}

Step 10: Solve for the Angle

To solve for the angle, we can use the following trigonometric identity:

sin⁑2(x)+cos⁑2(x)=1\sin^2(x) + \cos^2(x) = 1

We can substitute this expression into the equation above:

3=1+3sin⁑2(x)cos⁑2(x)3 = 1 + 3 \frac{\sin^2(x)}{\cos^2(x)}

Step 11: Solve for the Angle

To solve for the angle, we can subtract 11 from both sides of the equation:

2=3sin⁑2(x)cos⁑2(x)2 = 3 \frac{\sin^2(x)}{\cos^2(x)}

Step 12: Solve for the Angle

To solve for the angle, we can divide both sides of the equation by 33:

23=sin⁑2(x)cos⁑2(x)\frac{2}{3} = \frac{\sin^2(x)}{\cos^2(x)}

Step 13: Solve for the Angle

To solve for the angle, we can take the square root of both sides of the equation:

23=sin⁑(x)cos⁑(x)\sqrt{\frac{2}{3}} = \frac{\sin(x)}{\cos(x)}

Step 14: Solve for the Angle

To solve for the angle, we can use the following trigonometric identity:

tan⁑(x)=sin⁑(x)cos⁑(x)\tan(x) = \frac{\sin(x)}{\cos(x)}

We can substitute this expression into the equation above:

23=tan⁑(x)\sqrt{\frac{2}{3}} = \tan(x)

Step 15: Solve for the Angle

To solve for the angle, we can use the inverse tangent function:

x=arctan⁑(23)x = \arctan(\sqrt{\frac{2}{3}})

Step 16: Simplify the Expression

We can simplify the expression above by rewriting it as:

x=arctan⁑(23)x = \arctan\left(\sqrt{\frac{2}{3}}\right)

Step 17: Evaluate the Expression

We can evaluate the expression above by using a calculator:

xβ‰ˆ0.588x \approx 0.588

Step 18: Convert the Angle to Degrees

We can convert the angle to degrees by multiplying it by 180Ο€\frac{180}{\pi}:

xβ‰ˆ33.69∘x \approx 33.69^\circ

Step 19: Simplify the Expression

We can simplify the expression above by rewriting it as:

xβ‰ˆ33.69∘x \approx 33.69^\circ

Step 20: Evaluate the Expression

We can evaluate the expression above by using a calculator:

xβ‰ˆ33.69∘x \approx 33.69^\circ

Conclusion

Q: What is the inverse cotangent function?

A: The inverse cotangent function, denoted as arccot⁑(x)\operatorname{arccot}(x), is the inverse of the cotangent function. It returns the angle whose cotangent is a given value.

Q: What is the range of the inverse cotangent function?

A: The range of the inverse cotangent function is (βˆ’Ο€2,Ο€2)(-\frac{\pi}{2}, \frac{\pi}{2}), which means that it returns an angle between βˆ’Ο€2-\frac{\pi}{2} and Ο€2\frac{\pi}{2} radians.

Q: How do I solve an equation involving the inverse cotangent function?

A: To solve an equation involving the inverse cotangent function, you need to find the angle whose cotangent is equal to the given value. You can use trigonometric identities and inverse trigonometric functions to solve for the angle.

Q: What is the difference between the inverse cotangent function and the inverse tangent function?

A: The inverse cotangent function and the inverse tangent function are both inverse trigonometric functions, but they return different values. The inverse cotangent function returns the angle whose cotangent is a given value, while the inverse tangent function returns the angle whose tangent is a given value.

Q: Can I use a calculator to solve an equation involving the inverse cotangent function?

A: Yes, you can use a calculator to solve an equation involving the inverse cotangent function. Most calculators have a built-in inverse cotangent function that you can use to find the angle.

Q: What is the relationship between the inverse cotangent function and the cotangent function?

A: The inverse cotangent function is the inverse of the cotangent function. This means that if you take the cotangent of an angle and then take the inverse cotangent of that value, you will get the original angle back.

Q: Can I use the inverse cotangent function to solve a problem involving a right triangle?

A: Yes, you can use the inverse cotangent function to solve a problem involving a right triangle. The inverse cotangent function can be used to find the angle of a right triangle when you know the lengths of the sides.

Q: What are some common applications of the inverse cotangent function?

A: The inverse cotangent function has many applications in mathematics, physics, and engineering. Some common applications include solving right triangle problems, finding the angle of a rotating object, and calculating the height of a building.

Q: Can I use the inverse cotangent function to solve a problem involving a circular motion?

A: Yes, you can use the inverse cotangent function to solve a problem involving a circular motion. The inverse cotangent function can be used to find the angle of a circular motion when you know the radius and the angular velocity.

Q: What are some common mistakes to avoid when using the inverse cotangent function?

A: Some common mistakes to avoid when using the inverse cotangent function include:

  • Not checking the domain of the function
  • Not checking the range of the function
  • Not using the correct units
  • Not checking for extraneous solutions

Q: Can I use the inverse cotangent function to solve a problem involving a parametric equation?

A: Yes, you can use the inverse cotangent function to solve a problem involving a parametric equation. The inverse cotangent function can be used to find the angle of a parametric equation when you know the parametric equations.

Conclusion

In this article, we answered some frequently asked questions about the inverse cotangent function. We covered topics such as the definition of the inverse cotangent function, its range, and its applications. We also discussed some common mistakes to avoid when using the inverse cotangent function.