Find all values of x, 0<x<(ƒÎ/2) for which (1/�ã3)<cotx<(�ã3)
To find the values of x that satisfy the given inequality (1/√3) < cot(x) < (√3), we need to consider the unit circle and the behavior of the cot(x) function within the specified range.
Let's break the problem down step by step:
Step 1: Understand cotangent function:
The cotangent function is defined as cot(x) = cos(x) / sin(x). It represents the ratio of the adjacent side to the opposite side in a right triangle.
Step 2: Convert the inequality:
Rewriting the given inequality in terms of sine and cosine:
(1/√3) < cot(x) < (√3)
=> (1/√3) < cos(x) / sin(x) < (√3)
=> sin(x)/(√3) < cos(x) < (√3)sin(x)
Step 3: Analyze the unit circle:
The values of sin(x) and cos(x) change as x varies between 0 and π/2 on the unit circle. We need to examine the behavior of the cosine function within this range.
When x is between 0 and π/2, sin(x) is positive, and cos(x) is also positive. Therefore, we can multiply the inequality by sin(x) without changing the direction:
sin(x) / (√3) < cos(x) < (√3)sin(x)
Step 4: Simplify the inequality:
To simplify the inequality, consider the two sides separately:
- For the left side, sin(x) / (√3) < cos(x):
Multiply both sides by √3 to get:
sin(x) < √3 * cos(x)
- For the right side, cos(x) < (√3)sin(x):
Divide both sides by sin(x):
cos(x)/sin(x) < √3
Step 5: Combine the results:
Combining both results, we have:
sin(x) < √3 * cos(x) and cos(x)/sin(x) < √3
Step 6: Identify the values of x:
To find the values of x, we need to consider where sin(x) < √3 * cos(x) and cos(x)/sin(x) < √3 simultaneously.
Within the given range 0 < x < (π/2), the angles that satisfy these conditions occur in the first quadrant of the unit circle and also slightly before π/4 and slightly after 3π/4.
So, the values of x are:
- (π/6, π/4)
- (3π/4, 5π/6)
Note: The given inequality only holds true within 0 < x < (π/2). If you are looking for solutions in a different range, repeat the analysis for that range.