Find all values of x, 0<x<(pi/2) for which (1/( root 3))<cotx<(root 3)
To find all values of x such that (1/√3) < cot(x) < √3, we can use the properties of the cotangent function and build an approach step-by-step.
1. First, recall that cot(x) = cos(x) / sin(x) and 1/√3 is approximately equal to 0.5773, while √3 is approximately equal to 1.732.
2. Next, we can rewrite the given inequality as 0.5773 < cos(x) / sin(x) < 1.732.
3. To simplify further, we can multiply all parts of the inequality by sin(x) to get 0.5773sin(x) < cos(x) < 1.732sin(x).
4. Now, we are left with a single trigonometric function, cos(x), on both sides of the inequality sign.
5. Since we are interested in values of x between 0 and π/2, we need to analyze the behavior of cos(x) and sin(x) in this interval.
6. We know that sin(x) is positive and increasing from 0 to π/2, while cos(x) is positive and decreasing in the same interval. Therefore, we can take advantage of this information to solve the inequality.
7. First, let's work with the left side of the inequality: 0.5773sin(x) < cos(x).
8. Divide both sides by cos(x) (since cos(x) > 0), and we have 0.5773tan(x) < 1.
9. Next, recall that tan(x) = sin(x) / cos(x). We can substitute this in the previous equation: 0.5773sin(x) / cos(x) < 1.
10. Rearrange the terms: sin(x) < 1.732cos(x).
11. Now, let's work with the right side of the inequality: cos(x) < 1.732sin(x).
12. Similarly, divide both sides by sin(x) (since sin(x) > 0), and we have cos(x) / sin(x) < 1.732.
13. Substitute cot(x) = 1/tan(x) = cos(x) / sin(x), and we get cot(x) < 1.732.
14. Combining both inequalities, we have 0.5773tan(x) < 1 and cot(x) < 1.732.
15. To solve 0.5773tan(x) < 1, divide both sides by 0.5773 to get tan(x) < 1.7324.
16. Using a calculator or trigonometric table, find the range of values of x in the interval 0 < x < π/2 for which tan(x) is less than 1.7324.
17. The values of x that satisfy the inequality tan(x) < 1.7324 will also satisfy the inequality cot(x) < √3.
Therefore, by finding the values of x that satisfy the inequality tan(x) < 1.7324, we can determine all the values of x in the given interval for which (1/√3) < cot(x) < √3.