Evaluate. Write your answer in simplified, rationalized form. Do not round.
cot 45°
We know that cotangent (cot) is the reciprocal of tangent (tan). The tangent of 45° is equal to 1, so the cotangent of 45° is equal to the reciprocal of 1, which is 1/1 or simply 1. Thus, the simplified, rationalized form of cot 45° is 1.
To find the value of cotangent 45°, we need to find the reciprocal of the tangent of 45°. The tangent of 45° is equal to 1, so the cotangent of 45° is 1/1 or simply 1.
To evaluate cot 45°, we can use the trigonometric identity:
cot θ = 1 / tan θ
First, let's find the value of tan 45°:
Tan θ is equal to the ratio of the opposite side to the adjacent side in a right triangle. In a 45°-45°-90° triangle, the opposite side and adjacent side have equal length.
Let's assume the length of the opposite side and adjacent side is equal to x. According to the Pythagorean theorem, the hypotenuse is √2 times the length of each leg.
Using the Pythagorean theorem: x^2 + x^2 = (sqrt(2)x)^2
2x^2 = 2x^2
x^2 = x^2
Taking the square root of both sides:
√(x^2) = √(x^2)
x = x
Therefore, the opposite side, adjacent side, and hypotenuse in a 45°-45°-90° triangle are all equal.
Now, let's calculate the value of tan 45°:
tan 45° = opposite/adjacent = x/x = 1
Using the trigonometric identity cot θ = 1/tan θ, we can find cot 45°:
cot 45° = 1/tan 45° = 1/1 = 1
So, the value of cot 45° is 1 in simplified, rationalized form.