An angle of rotation describes how much a ray rotates around its endpoint. It consists of three key components: the rotation center or vertex, the initial ray called the initial side, and the final ray called the terminal side. When the ray rotates from the initial position to the terminal position, the angle formed is the angle of rotation.
The direction of rotation determines the sign of the angle. Counterclockwise rotation produces positive angles, while clockwise rotation produces negative angles. This follows the standard mathematical convention used in coordinate systems. For example, rotating 90 degrees counterclockwise gives us positive π/2, while rotating 60 degrees clockwise gives us negative π/3.
Angles can be measured in two main units: degrees and radians. In degrees, a quarter turn is 90 degrees, a half turn is 180 degrees, three-quarters is 270 degrees, and a full rotation is 360 degrees. In radians, these same angles are π/2, π, 3π/2, and 2π respectively. The conversion formula is: radians equals degrees times π divided by 180 degrees.
Coterminal angles are angles that have the same terminal side but different measures. To find coterminal angles, we add or subtract multiples of 360 degrees or 2π radians. For example, 30°, 390°, and -330° are all coterminal because they all end at the same position when rotated from the initial side.
Rotation angles have many practical applications. They're used in circular motion and wheel rotations, clock hands for time measurement, navigation with compass bearings, computer graphics and animations, trigonometry and wave functions, and various engineering and mechanical systems. Understanding rotation angles is fundamental to many areas of mathematics and science.
旋转方向决定角度的正负性。逆时针旋转产生正角，顺时针旋转产生负角。这是数学中使用的标准约定。让我们看看一些常见角度的示例：正90度是逆时针四分之一圈，负90度是顺时针四分之一圈，正180度是逆时针半圈，负180度是顺时针半圈。
角度有两种主要的测量单位：角度制和弧度制。角度制以度为单位，一个完整圆周是360度，常用于日常生活。弧度制以弧度为单位，一个完整圆周是2π弧度，在数学中更常用。两者的换算关系是：π弧度等于180度。让我们看看一些常见角度的对应关系：0度对应0弧度，90度对应π/2弧度，180度对应π弧度，270度对应3π/2弧度。
终边相同角是指相差360度整数倍的角，它们具有相同的终边位置。数学表达式为α加k乘以360度，其中k是整数。例如，30度、390度和负330度都有相同的终边。让我们通过动画来观察：从30度开始，加上360度得到390度，再加360度得到750度，减去360度得到负330度，它们的终边都指向同一个方向，这就是旋转角的周期性质。
象限角是根据终边所在象限进行分类的角度。第一象限角的范围是0度到90度之间，第二象限角是90度到180度之间，第三象限角是180度到270度之间，第四象限角是270度到360度之间。还有特殊的轴线角：x轴正半轴对应0度和360度，y轴正半轴对应90度，x轴负半轴对应180度，y轴负半轴对应270度。让我们通过动画来观察不同角度对应的象限。