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
Point F lies on a unit circle in the xy-plane and has coordinates \( \left(1,0\right) \) . Point G is the center of the circle and has coordinates \( (0,0). \) Point H also lies on the circle and has coordinates\( \left(-1,\ y\right) \) , where \( y \) is a constant. Which of the following could be the positive measure of angle FGH, in radians?
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三角函数・单位圆与弧度进阶
If cos A = 0.69, what is the value of sin C?
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三角函数・单位圆与弧度进阶
The measure of angle B is \( \frac{7π}{4} \) radians. What is the value of cos(B)?
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三角函数・单位圆与弧度进阶
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
三角函数・单位圆与弧度进阶
The measure of angle \( C \) is \( \frac{3π}{4} \) radians. What is the value of \( cos(C) \)?
原库参考解析
三角函数・单位圆与弧度挑战
In right triangle ABC, angles A and B are acute, side AC has a length of 28.2, and \( tanB=\frac{1}{3} \). What is the length of side BC rounded to the nearest tenth?
原库参考解析
三角函数・单位圆与弧度进阶
The difference between the measure of angle A and the measure of angle B is \( −\frac{5}{12}π \) radians. Which expression shows the difference between the measure of angle A and the measure of angle B, in degrees?
原库参考解析
三角函数・单位圆与弧度进阶
The measure of angle \( B \) is \( \frac{3π}{4} \) radians. What is the value of \( sin(B) \) ?
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三角函数・单位圆与弧度进阶
In triangle \( ABC \), \( BC=36\sqrt{3} \). What is the value of \( cosA \) ?
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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 22.7 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
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 29.1 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 right triangle ABC, side BC has a length of \( \sqrt{799} \) . The hypotenuse, side AB has a length of 32. What is the value of \( sinB \) ?
原库参考解析
