这部分需要会什么？
- 按指定角识别正弦、余弦和正切
- 使用互余关系与单位圆坐标
- 在角度与弧度之间换算并计算弧长
核心概念与公式
直角三角形中，对锐角 \(\theta\)，\(\sin\theta=\text{对边}/\text{斜边}\)，\(\cos\theta=\text{邻边}/\text{斜边}\)，\(\tan\theta=\text{对边}/\text{邻边}\)。互余角满足 \(\sin\theta=\cos(90^\circ-\theta)\)。
单位圆上角 \(\theta\) 对应点为 \((\cos\theta,\sin\theta)\)，所以 \(\sin^2\theta+\cos^2\theta=1\)。\(180^\circ=\pi\) 弧度；用角度乘 \(\pi/180\) 得弧度。弧长公式 \(s=r\theta\) 要求 \(\theta\) 为弧度。
做题时怎样使用
- 以题目指定的角为参照，重新辨认对边与邻边
- 根据已知边和所求边选择三角比
- 代入前核对角度单位；涉及单位圆时检查象限和符号
容易混淆的地方
换一个锐角，对边和邻边会交换。\(s=r\theta\) 中不能直接代入角度数，例如 60 必须先换成 \(\pi/3\)。
用一个例题串起方法
知识点教学例题半径为 6 的圆中，圆心角为 \(60^\circ\) 所对的劣弧长是多少？
展开解题思路 ＋
\(60^\circ=\pi/3\) 弧度，所以 \(s=6\times\pi/3=2\pi\)。也可用周长的 \(60/360\) 来计算。
本知识点题库
先独立作答，再查看答案与解析。
三角函数・单位圆与弧度进阶
What is the value of \( tanX \) in the triangle shown?
原库参考解析
三角函数・单位圆与弧度挑战
In triangle JKL, the measure of angle J is 90b°, the measure of angle K is 66a°, and the measure of angle L is 24a°, where a and b are constants. Which of the following must be true?
原库参考解析
三角函数・单位圆与弧度进阶
The measure of angle G is \( \frac{π}{9} \) radians. If the measure of angle G is \( 2n \) degrees, where n is a constant, what is the value of n?
原库参考解析
三角函数・单位圆与弧度进阶
In triangles \( RST \) and \( XYZ \) shown, \( XY \) is parallel to \( TS \) and \( tanR=\frac{120}{391} \) . What is the value of \( sinX \) in triangle \( XYZ \)?
原库参考解析
三角函数・单位圆与弧度进阶
In triangle \( QRS \) shown, \( QR<RS \). The length of one side is 22. Which expression represents the length of \( QS \) ?
参考答案
三角函数・单位圆与弧度进阶
Which measure, in radians, is equivalent to \( (47·180)° \)?
参考答案
三角函数・单位圆与弧度进阶
In the right triangle shown, a, b and c are positive constants. For an acute angle D, \( sinD=\frac{b}{c} \). What is the measure of angle D, in degrees?
参考答案
三角函数・单位圆与弧度挑战
In triangle RST, RS=ST and the length of \( RT \) is 76 units. If \( tanR=\frac{9}{38} \), what is the area, in square units, of triangle RST?
参考答案
三角函数・单位圆与弧度挑战
In the figure, \( AD \) and \( BC \) intersect at point D. The measure of \( ∠ABD \) is w°, the measure of \( ∠CAD \) is z°, and AD=143. If \( cosw°=\frac{9}{41} \), what is the value of \( tanz° \)?
参考答案
三角函数・单位圆与弧度进阶
In triangle LMN, the measure of angle L is 90°, MN=105, and LM=2. What is the value of \( tanM \)?
参考答案
三角函数・单位圆与弧度挑战
Which measure, in radians, is equivalent to (47⋅180)°?
参考答案
三角函数・单位圆与弧度进阶
In triangle PQR, the sine of ∠Q is \( \frac{1}{2} \). What is the length of PQ?
参考答案
