Solve the equation cotx=4 in the interval 0<(smaller and equal to)x<<(larger and equal to)2pie
If cot x =- 4, tan x = 1/4.
The tan function is positive in the first and thrid quadrants.
In the first quadrant, the angle would be arctan 0.25 = 14.04 degrees. There is another solution in the third quadrant. See if you can figure it out.
To solve the equation cot(x) = 4 in the given interval, we need to find the values of x that satisfy this equation. Let's break down the steps to solve it:
Step 1: Understanding cotangent and its range
Cotangent (cot) is the reciprocal of the tangent function. It can also be expressed as the ratio of the adjacent side to the opposite side of a right triangle. The cotangent function has a period of π (pi), meaning its values repeat every interval of π units.
Step 2: Recognizing the equation
The equation cot(x) = 4 means that the ratio of the adjacent side to the opposite side of a right triangle, when x is the angle, is equal to 4.
Step 3: Finding the angle in the given interval
Since we are given the interval 0 ≤ x ≤ 2π (0 to 2π), we need to identify the angles within this range that satisfy the equation. Let's list down the angles that correspond to the cotangent of 4:
- First, find the angle whose cotangent is 4 by taking the inverse cotangent (also known as arccot) of 4: x = arccot(4).
- Use a calculator to find the value of the inverse cotangent of 4, which is approximately 0.24498 radians or about 14.04 degrees.
- Since cotangent has a period of π, we need to add any multiple of π to the angle we found to get other solutions within the given interval. In this case, we add multiples of π to 0.24498 radians.
Step 4: Expressing the solutions
Since the given interval is 0 ≤ x ≤ 2π, we need to express the solutions in terms of x that fall within this range. Here's the list of solutions:
- The initial solution is x = 0.24498 radians (approximately).
- Adding multiples of π, we can express the solutions within the given interval as:
- x = 0.24498 + π ≈ 3.3862 radians (approximately),
- x = 0.24498 + 2π ≈ 6.4283 radians (approximately).
Therefore, the solutions to the equation cot(x) = 4 in the interval 0 ≤ x ≤ 2π are approximately x ≈ 0.24498 radians, x ≈ 3.3862 radians, and x ≈ 6.4283 radians.