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
Triangle FGH is similar to triangle JKL, where angle F corresponds toangle J and angles G and K are right angles.If sin(F) = \( \frac{308}{317} \) , what is the value of sin(J)?
原库参考解析
三角函数・单位圆与弧度基础
An angle has a measure of
radians. What is the measure of the angle, in degrees?原库参考解析
三角函数・单位圆与弧度挑战
An angle has a measure of
radians. What is the measure of the angle, in degrees?原库参考解析
三角函数・单位圆与弧度进阶
The measure of angle R is
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
. 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
RS = 20
ST = 48
TR = 52
The side lengths of right triangle RST are given. Triangle RST is similar to triangle UVW, where S corresponds to V and T corresponds to W. What is the value of tan W?
原库参考解析
三角函数・单位圆与弧度进阶
What is the value of\( tan\frac{92π}{3} \) ？
原库参考解析
三角函数・单位圆与弧度进阶
In the triangle shown, what is the value of cos x°?
原库参考解析
三角函数・单位圆与弧度进阶
Which of the following expressions is equivalent to (sin 24°)(cos 66°)+(cos 24°)(sin 66°)?
原库参考解析
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 \( \frac{\sqrt{15}}{4} \). 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
Triangle \( FGH \) is similar to triangle \( JKL \), where angle \( F \) corresponds to angle \( J \) and angles \( G \) and \( K \) are right angles. If \( sin(F)=\frac{308}{317} \) , what is the value of \( sin(J) \)?
原库参考解析
三角函数・单位圆与弧度基础
The measure of angle \( R \) is \( \frac{2π}{3} \) radians. The measure of angle \( T \) is \( \frac{5π}{12} \) radians greater than the measure of angle \( R \). What is the measure of angle \( T \), in degrees?
原库参考解析
