If theta� is a standard position angle and cot �theta = 1.85, in which quadrant(s) could �theta lie?
To determine in which quadrant(s) theta could lie when cot(theta) = 1.85, we need to understand the relationship between the tangent function and the quadrants on the unit circle.
1. Start with the equation cot(theta) = 1.85. The cotangent (cot) function is the reciprocal of the tangent (tan) function, so we can rewrite it as tan(theta) = 1/1.85.
2. Recall that the tangent function is positive in the first and third quadrants. Therefore, the values of theta could lie in either the first or third quadrant.
3. To determine the specific quadrant, we need to consider the value of tan(theta). Given tan(theta) = 1/1.85, we can determine the approximate value of theta.
4. Calculate the arctan(1/1.85) using a calculator, which gives approximately 28.96 degrees.
5. Since theta is a standard position angle, it measures the counterclockwise angle from the positive x-axis to the terminal side of the angle. In the first quadrant, theta is measured from 0 to 90 degrees. In the third quadrant, theta is measured from -90 to 0 degrees.
6. The calculated angle of approximately 28.96 degrees falls within the first quadrant (from 0 to 90 degrees). Therefore, theta could lie in the first quadrant.
In conclusion, based on the equation cot(theta) = 1.85, theta could lie in the first quadrant.