Solve the equations to the nearest tenth and use the given restrictions.
2. sin theta = -0.204, for 90 degs < theta < 270 degs
Ans: 191.8 degs?
To solve the equation sin(theta) = -0.204 for the given restrictions (90° < theta < 270°), we need to find the angle theta that satisfies this equation within the given range.
First, we need to find the reference angle, which is the angle in the first quadrant that has the same sine value as the given angle. Since the sine function is negative for angles in the 2nd and 3rd quadrants, the reference angle will be the angle in the first quadrant with the same absolute value of the sine.
To find the reference angle, we can take the inverse sine (also known as arcsine) of the absolute value of the given sine value:
Reference angle = arcsin(|-0.204|) ≈ 11.8°
Since the given angle is in the 2nd and 3rd quadrants, we have two solutions: one in the 2nd quadrant and another in the 3rd quadrant. To find these solutions, we subtract the reference angle from 180° for the 2nd quadrant solution and add the reference angle to 180° for the 3rd quadrant solution.
2nd quadrant solution:
theta = 180° - 11.8° ≈ 168.2°
3rd quadrant solution:
theta = 180° + 11.8° ≈ 191.8°
Therefore, the solutions to the equation sin(theta) = -0.204 for the given restrictions (90° < theta < 270°) are approximately 168.2° and 191.8°. So, your answer of 191.8° is correct.