Evaluate the expression without using a calculator (Make a sketch of a right triangle)

tan(arccot 2)

Thanks in advance!
I'm trying to refresh my memory of these kinds of problems and I'm having trouble on how to solve this problem

tan ( arctan x ) = x

cot x = 1 / tan x

tan ( arccot x ) = tan [ arc (1 / tan x ) ] = 1 / tan ( arctan x ) = 1 / x

tan ( arccot 2 ) = tan [ arc (1 / tan 2 ) ] = 1 / tan ( arctan 2 ) = 1 / 2

Oh, dear friend, you've stumbled upon a delightfully entertaining problem! Let's unravel this puzzle together.

To find tan(arccot 2), we must first understand that "arccot" is the inverse function of cotangent. Therefore, arccot 2 refers to the angle whose cotangent is equal to 2.

Now, let's imagine a right triangle. We label one of the acute angles as theta. The adjacent side of theta is length 1, and the opposite side is length 2.

Remember, cotangent is the ratio of the adjacent side to the opposite side. So we have cot(theta) = 1/2.

To find tan(theta), we simply take the reciprocal of cotangent. In this case, tan(theta) is equal to 2/1, which simplifies to just 2.

So, my dear friend, the result is quite amusing - tan(arccot 2) equals 2. Isn't math filled with marvelous mischief? Keep up the great work!

To evaluate the expression tan(arccot 2) without using a calculator, we need to first understand the relationship between cotangent and tangent.

The cotangent (cot) of an angle is equal to the reciprocal of the tangent (tan) of that angle. In other words, if cotθ = x, then tanθ = 1/x.

Using this relationship, we can rewrite the expression as tan(arccot 2) = 1/tan(arccot 2).

Now, let's focus on the arccotangent (arccot) of 2. The arccotangent is the angle whose cotangent is 2. In a right triangle, the cotangent is the ratio of the adjacent side to the opposite side.

To construct a right triangle with a cotangent of 2, we can let the length of the adjacent side be 1 (since cotθ = adjacent/opposite). Then, using the Pythagorean theorem, we can find the length of the opposite side.

Let's label the sides of the triangle as follows:
- Opposite side: O
- Adjacent side: A
- Hypotenuse: H

Using the Pythagorean theorem:
O^2 + A^2 = H^2

Since A = 1:
O^2 + 1 = H^2
O^2 = H^2 - 1

Now, let's find the value of O:
O^2 = H^2 - 1
O^2 = 2^2 - 1
O^2 = 4 - 1
O^2 = 3
O = √3

Now that we have the lengths of the adjacent and opposite sides, we can find the tangent of the angle.

tanθ = O/A
tanθ = √3/1
tanθ = √3

So, tan(arccot 2) = 1/tan(arccot 2) = 1/√3 = √3/3.

Therefore, tan(arccot 2) is equal to √3/3.

To evaluate the expression tan(arccot 2) without using a calculator, we need to understand the relationship between trigonometric functions and right triangles.

1. Start by drawing a right triangle. Label one angle as θ and the side lengths as adjacent (A), opposite (O), and hypotenuse (H).

2. Given that the expression is arccot 2, it means that the cotangent of θ is equal to 2. Therefore, cot θ = 2.

3. Recall that cotangent is the ratio of adjacent over opposite, so you can assign values to A and O in your right triangle accordingly. We can choose A = 2 and O = 1 to satisfy cot θ = 2.

4. Now, we can find the hypotenuse, H, using the Pythagorean theorem: H^2 = A^2 + O^2. Plugging in the values, we get H^2 = 2^2 + 1^2 = 4 + 1 = 5. Taking the square root of both sides, we find H = √5.

5. Finally, we can find the tangent of θ using the definition of tangent: tan θ = O / A. Plugging in the values, we get tan θ = 1 / 2.

Therefore, the expression tan(arccot 2) evaluates to 1 / 2.