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
In the right triangle shown, the value of tan x° is \( \frac{c}{39} \). What is the value of c?
原库参考解析
三角函数・单位圆与弧度进阶
Triangle \( XYZ \) shown is a right triangle. Which of the following has the same value as sin \( X \)?
原库参考解析
三角函数・单位圆与弧度挑战
In right triangle RST, the sum of the measures of angle R and angle S is 90 degrees. The value of sin(R) is \( \frac{3\sqrt{17}}{13} \). What is the value of cos(S)?
原库参考解析
三角函数・单位圆与弧度进阶
The measure of angle R is \( \frac{5}{6}π \) radians. The measure of angle T is \( \frac{7π}{12} \)radians greater than the measure of angle R. What is the measure of angle T, in degrees?
原库参考解析
三角函数・单位圆与弧度进阶
According to a US law, ramps for use by the general public must form an angle with level ground such that \( tanθ≤\frac{1}{12} \). If the ramp in the figure conforms to this law and has a height of 28.4 inches, what is the least possible value of x, in inches?
原库参考解析
三角函数・单位圆与弧度进阶
In the triangle shown, \( RS=\sqrt{167} \). What is the value of sin \( R \)?
原库参考解析
三角函数・单位圆与弧度进阶
What is the value of sin 18π?
原库参考解析
三角函数・单位圆与弧度进阶
In triangle \( QRS \) shown, \( QR<RS \) . Which expression represents the length of \( QS \) ?
原库参考解析
三角函数・单位圆与弧度进阶
In the right triangle shown, c is a constant. The value of \( sinx°=\frac{k}{6} \) where k is a constant. What is the value of k?
原库参考解析
三角函数・单位圆与弧度进阶
In the right triangle shown, c is a constant. The value of \( sinx°=\frac{k}{c} \), where k is a constant. What is the value of k?
原库参考解析
三角函数・单位圆与弧度挑战
In right triangle \( JKL \), the tangent of \( ∠L \) is \( \frac{3}{4} \). What is the length of \( JK \) ?
原库参考解析
三角函数・单位圆与弧度挑战
In right triangle RST, the sum of the measures of angle R and angle S is 90 degrees. The value of sin(R) is \( \frac{2\sqrt{10}}{7} \). What is the value of cos(S)?
原库参考解析
