theta is an angle in standard position with a domain of 0 degrees < theta < 350 degrees and cot theta = -(24)/(7). Find all possible values of theta (accurate to 0.1 degree if necessary).
cot < 0 in QII and QIV
cot 16.26° = 24/7, so
theta = 180-16.26 and 360-16.26
To find all possible values of theta, we need to determine the angles in standard position that satisfy the equation cot(theta) = -(24/7).
First, let's recall the definition of the cotangent function. The cotangent of an angle is the ratio of the adjacent side to the opposite side in a right triangle. It can also be expressed as the reciprocal of the tangent function: cot(theta) = 1/tan(theta).
Since cot(theta) = -(24/7), we can rewrite it as 1/tan(theta) = -(24/7).
Now, we can find the inverse tangent of both sides to solve for theta. Take note that the range of the inverse tangent function is -π/2 < theta < π/2, or -90° < theta < 90°. However, our given domain is 0° < theta < 350°.
To accommodate the given domain, we can add a full revolution to the angles that fall within the range -90° < theta < 0°. Adding 360° to these angles will bring them into the range we are looking for.
Therefore, the steps to find all possible values of theta are as follows:
1. Calculate the inverse tangent of -(24/7) using a calculator:
tan^(-1)(-(24/7)) ≈ -77.5° (to the nearest tenth)
2. Since the result is in the range -90° < theta < 0°, add 360° to obtain a solution within the given domain:
-77.5° + 360° ≈ 282.5° (to the nearest tenth)
Thus, one possible value of theta that satisfies the equation is approximately 282.5°.