Question

Use the circle shown in the rectangular coordinate system to find two angles, in radians, between -2x and 2x such that

each angle's terminal side passes through the origin and the point indicated on the circle.
Use a comma to separate answers as needed. (Photo attached).

Answer 1

To find angles passing through a given point on a circle in a rectangular coordinate system, polar coordinates are used. The angle is found using the inverse tangent function and considering the quadrant the point lies in. Angles are positive in the counter-clockwise direction, and conversion between degrees and radians is achieved using π/180.

Step-by-step explanation:

To find the angles in radians such that each angle's terminal side passes through the origin and a given point on the circle in a rectangular coordinate system, we typically use polar coordinates. In polar coordinates, the position of a point is determined by the radial distance from the origin and the angle from the positive x-axis.

Firstly, we identify the quadrant in which the point on the circle lies. Then we can use the inverse tangent function, commonly written as tan-1, to find the direction angle. Since angles are defined as positive in the counter-clockwise direction, angles in the first and second quadrants will be positive, while those in the third and fourth quadrants will be negative when measured from the positive x-axis.

To convert an angle from degrees to radians, we use the conversion factor π radians = 180 degrees. Therefore, an angle in degrees can be converted to radians by multiplying by the factor π/180.