Using the standard restricted domain for the cotangent function, which of the following best describes the behavior of the inverse cotangent function?
-a decreasing function defined in quadrants I and II
-a decreasing function defined in quadrants III and IV
-an increasing function defined in quadrants I and II
-an increasing function defined in quadrants III and IV
10. Is a decreasing function defined in quadrants 1 and 2
11. Is 51°
A) a decreasing function defined in Quadrants I and II
1. C (28.3)
2. E (sec x)
3. B (tan x and sec x)
4. E (sec x)
5. A (2.5 seconds)
6. D (a=1; b=-1; c=pi)
7. C (f(x)=2sin(-1.5x+0.5)-2)
8. B (theta(t)=6cos(2pi/3 t)
9. C (x=[20,30])
10. A (a decreasing function defined in Quadrants I and II)
11. B (51°)
12. D (sqrt 1-x^2)
13. E (arccos (cosx)=x)
14. D (10.6 mins)
honors student chad
Well, let's try to unravel the mystery of the inverse cotangent function, shall we?
You see, the standard restricted domain for the cotangent function is the interval between -π/2 and π/2. This means that the cotangent function repeats its values every π units, like a truly mesmerizing carousel ride.
Now, when we take a look at the inverse cotangent function, it's like hopping on the carousel in reverse. The domain of the inverse cotangent function is the entire set of real numbers.
Here comes the funny part, so hold on tight! Since the cotangent function is decreasing in quadrants I and IV, its inverse, the cotangent inverse function, is increasing in those quadrants. You know, it's the opposite, like trying to dance the Macarena while doing the Moonwalk. It's a whole new level of coordination!
So, my dear question-asker, the answer is: An increasing function defined in quadrants I and II. Cheers to the quirky world of inverse trigonometric functions!
To determine the behavior of the inverse cotangent function, we must first understand the behavior of the cotangent function itself.
The cotangent function (cot(x)) is defined as the reciprocal of the tangent function (tan(x)), where cot(x) = 1/tan(x). It signifies the ratio of the adjacent side to the opposite side in a right triangle.
The cotangent function has vertical asymptotes at x = π/2, 3π/2, 5π/2, etc. These points are excluded from the domain of the cotangent function. Hence, the standard restricted domain for cot(x) is (-∞,π/2) U (π/2, 3π/2) U (3π/2,5π/2) U ....
Now, the behavior of the inverse cotangent function (arccot(x) or cot^(-1)(x)) is the inverse of the cotangent function. It takes the output of the cotangent function as an input and returns the corresponding angle.
Considering the standard restricted domain for the cotangent function, the inverse function will have a restricted range of (0,π). It means that the output of the inverse cotangent function will be an angle between 0 and π.
In the coordinate plane, quadrants I and II have angles between 0 and π, while quadrants III and IV have angles between π and 2π.
Therefore, based on the restricted range of the inverse cotangent function, the correct answer is:
- An increasing function defined in quadrants I and II.