Trigonometry

General Angles and Radian Measure

1. General Angles
1. Angle and Its Measure
   In a plane, when the ray rotates from the initial ray around point , the figure formed by the two rays and is denoted by the symbol , and the amount of rotation is called the measure of . In this case, the ray is called the initial side, and the ray is called the terminal side. When the terminal side rotates around point , the counterclockwise direction is defined as the positive direction, and the clockwise direction as the negative direction. The angle's measure is expressed with a positive sign when rotating in the positive direction and a negative sign when rotating in the negative direction.
   
   
   
   
   
2. General Angle
   When an angle is given by the initial side and terminal side , if the measure of one of the angles formed by is , the measure of can be expressed as: (where is an integer)
   This is called the general angle represented by the terminal side
   
   
3. Quadrant Angles
   In the coordinate plane, when the initial side is along the positive direction of the x-axis, the angles formed by the terminal side \text{OP} in the first, second, third, and fourth quadrants are referred to as the angles of the first, second, third, and fourth quadrants, respectively.
   
   
   


2. Radian Measure
1. Radian Measure
   In a circle with center and radius , if the length of the arc is equal to , the measure of the central angle is defined as 1 radian. This method of measuring angles is called radian measure. When expressing angles in radian measure, the unit "radian" is usually omitted.
   
   
   
   
   
2. Relationship Between Degrees and Radians
   Since the length of an arc is proportional to the size of the central angle, we have the proportion:
   
   Hence, radian , and radians.
   
   
3. Arc Length and Area of a Sector
   In a sector with radius and central angle (in radians), if the arc length is and the area is , both and are proportional to the central angle . Therefore, we have: