Given Tan(A) = 5 in Quadrant III and Sin(B) = ⅔ in Quadrant II, what is the Quadrant of A-B?
Tan A = 5(Q3), A = 78.7o. S. of W. = 180+78.7 = 258.7o CCW.
Sin B = 2/3(Q2), B = 41.8o N. of W. = 180-41.8 = 138.2o CCW.
A-B = 258.7 - 138.2 = 120.5o, Q2.
To determine the quadrant of A-B, we need to subtract the angle B from angle A, and understand the quadrants in which each angle lies.
First, let's start by understanding the given information:
1. Tan(A) = 5 in Quadrant III: This means that the tangent of angle A is positive (since tan is positive in Quadrant III) and equals 5.
2. Sin(B) = ⅔ in Quadrant II: This means that the sine of angle B is positive (since sin is positive in Quadrant II) and equals ⅔.
To find the quadrant of A-B, we need to consider the signs of the trigonometric functions involved. Since tan(A) = 5 and sin(B) = ⅔, we can determine the following:
- Tangent is positive in Quadrants I and III, so angle A could either be in Quadrant I or Quadrant III.
- Sine is positive in Quadrants I and II, so angle B could either be in Quadrant I or Quadrant II.
Now let's calculate the value of A-B:
To determine the value of A-B, we need to find the numerical value of angle A and angle B.
To find angle A:
Since Tan(A) = 5, we can use the inverse tangent function (Tan^(-1)) to find angle A. So we have:
A = Tan^(-1)(5)
To find angle B:
Since Sin(B) = ⅔, we can use the inverse sine function (Sin^(-1)) to find angle B. So we have:
B = Sin^(-1)(⅔)
To determine the signs of angles A and B, we consider their respective trigonometric functions:
- Tan(A) = 5 (positive) -> angle A is in Quadrant I or III.
- Sin(B) = ⅔ (positive) -> angle B is in Quadrant I or II.
Now, plug the calculated angles back into the equation A - B and determine the resulting quadrant.
A-B = Tan^(-1)(5) - Sin^(-1)(⅔)
Evaluate the value of this expression using a calculator or math software, and determine the resulting quadrant based on the signs of angles A and B:
- If A > B and both A and B are positive, the resulting angle (A-B) will be in Quadrant IV.
- If A > B and A is positive while B is negative, the resulting angle (A-B) will be in Quadrant I.
- If A < B and both A and B are positive, the resulting angle (A-B) will be in Quadrant III.
- If A < B and A is positive while B is negative, the resulting angle (A-B) will be in Quadrant II.
Based on the given information, it is not possible to determine the quadrant of A-B without evaluating A - B using their actual numerical values.