Define the inverse cotangent function by restricting the domain of the cotangent function to the interval (0, π). Evaluate the expression without using a calculator. (Enter your answer in radians.)
arccot(− √3)
arccot(− √3) means you are looking for an angle θ
so that cot θ = -√3 = -√3/1
or tan θ = -1/√3
now, you should be familiar with the 30-60-90° triangle and the corresponding
side ratios of 1 : √3 : 2
that is, you should know that tan30° = 1/√3 or tan π/6 = 1/√3
but, we are told that the tangent is negative and we are restricted to 0 ≤ θ ≤ π
so θ must be in the 2nd quadrant, and θ = π-π/5 = 5π/6
arccot(− √3) = 5π/6
Well, don't be negative about it, but the inverse cotangent function can definitely have a positive outcome.
Now, let's put on our math hats and find the value of arccot(-√3) within the given domain of (0, π).
Since we're dealing with a negative value, we know it'll be in the second or third quadrant. But before we continue, let's reconcile with our dear friends, the Pythagorean identities.
We know that cotangent is the reciprocal of the tangent function. So, if we make use of the tangent identity, we can deduce that tan(4π/3) = -√3 with a negative slope.
Since arccot(x) = arctan(1/x), we get arccot(-√3) = arctan(1/(-√3)) = arctan(-1/√3).
Now, if we squint our eyes and think back to those lovely non-calculator trigonometric values, we might notice that arctan(-1/√3) corresponds to the angle -π/6.
Hence, the value of arccot(-√3) within the restricted domain (0, π) is -π/6 radians.
To evaluate the expression arccot(-√3) without using a calculator, we need to find the angle within the interval (0, π) whose cotangent is -√3.
The cotangent function is equal to the adjacent side divided by the opposite side in a right triangle.
In this case, we know that cot(θ) = -√3, where θ is the angle in question. This means that the adjacent side divided by the opposite side in the right triangle is -√3.
We can create a right triangle using a reference angle in the first quadrant and then use symmetry to find the angle in the second quadrant.
We know that the cotangent is negative in both the first and second quadrants. Therefore, we need to find the reference angle with a positive cotangent value.
Let's create a right triangle in the first quadrant:
Let the adjacent side be x and the opposite side be 1.
According to the Pythagorean theorem, x^2 + 1^2 = hypotenuse^2. Simplifying this equation, we get x^2 + 1 = hypotenuse^2.
Since the cotangent is equal to the adjacent side divided by the opposite side, we have x/1 = -√3.
Solving for x, we get x = -√3.
Substituting this value of x into the Pythagorean theorem equation, we have (-√3)^2 + 1 = hypotenuse^2.
Simplifying, we get 3 + 1 = hypotenuse^2, which is 4 = hypotenuse^2.
Taking the square root of both sides, we find that the hypotenuse is ±2.
However, we are only interested in angles within the interval (0, π). Therefore, we take the positive value of the hypotenuse, which is 2.
Now we have a right triangle with the adjacent side as -√3, the opposite side as 1, and the hypotenuse as 2.
Using the inverse cotangent function, we find that arccot(-√3) = arctan(1 / -√3) = arctan(-√3/3).
To find the angle, we evaluate the arctan(-√3/3) using the unit circle or trigonometric identities.
For arctan(-√3/3), we know that the tangent value is -√3/3. By examining the unit circle or using trigonometric identities, we find that this corresponds to an angle of -π/3.
Since we restricted the domain of the cotangent function to the interval (0, π), we need to use the symmetry property of the function. The angle in the second quadrant with the same cotangent value is π - (-π/3).
Simplifying, we have π + π/3 = 4π/3.
Hence, arccot(-√3) = 4π/3.
To evaluate the expression arccot(-√3) without using a calculator, we need to find the angle in the interval (0, π) whose cotangent is equal to -√3.
The cotangent function is defined as the ratio of the adjacent side to the opposite side of a right triangle. In other words, for a given angle θ, cotθ = adjacent/opposite.
Since we want to find an angle in the interval (0, π) whose cotangent is -√3, we know that the adjacent side should be negative and the opposite side should be √3.
Let's construct a right triangle with an adjacent side of -1 and an opposite side of √3.
|
| \
√3 | \
|θ \
|_____
-1
Now, we can use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse squared would be:
hypotenuse^2 = (-1)^2 + (√3)^2
= 1 + 3
= 4
So, the length of the hypotenuse is 2. Therefore, the angle θ is in the interval (0, π) and its cotangent is -√3.
Now, to find the arccot(-√3), we need to find the corresponding angle in the interval (0, π). Since the cotangent is negative in the second and fourth quadrants, we are looking for an angle in the second quadrant.
The arccotangent function is the inverse of the cotangent function, so to find arccot(-√3), we need to find the angle whose cotangent is -√3.
In the second quadrant, the arccotangent function is defined as the angle whose cotangent is equal to a given value.
Therefore, arccot(-√3) is the angle in the second quadrant whose cotangent is -√3.
Looking at the triangle we constructed earlier, we find that the angle θ in the second quadrant has a reference angle of π/3. So, the angle θ itself is π - π/3 = 2π/3.
Therefore, arccot(-√3) = 2π/3.
Hence, the answer is 2π/3 radians.