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・单位圆与弧度挑战
The measure of an angle is \( (51)(\frac{k}{2})π \) radians, where k is a constant. What is the measure of this angle, in degrees?
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 \( \frac{4\sqrt{2}}{9} \) . What is the value of \( cos(S) \) ?
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 23.3, and \( tanB=\frac{1}{4} \) . What is the length of side BC, rounded to the nearest tenth?
原库参考解析
三角函数・单位圆与弧度挑战
In triangle \( QRS \) shown, \( QR<RS \) . Which expression represents the length of \( QS \) ?
原库参考解析
三角函数・单位圆与弧度挑战
In the xy-plane, a unit circle with center at the origin O contains point A with coordinates \( (1,\ 0) \) and point B with coordinates \( (\frac{5}{\sqrt{34}},\ -\frac{3}{\sqrt{34}}) \) . If the measure of angle AOB is p radians, what is the value of \( \frac{cosp}{sinp} \) ?
原库参考解析
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 \( AD=178 \) . If \( cosw°=\frac{9}{41} \) , what is the value of \( tanz° \) ?
原库参考解析
三角函数・单位圆与弧度挑战
In triangle \( XYZ \), the measure of angle \( X \) is 90°. Point \( W \) lies on segment \( YZ \), and segment \( WX \) is perpendicular to segment \( YZ \). The length of segment \( YW \) is 612, and the length of segment \( WX \) is 459. What is the value of \( tanZ \)?
原库参考解析
三角函数・单位圆与弧度挑战
Triangle \( ABC \) is similar to triangle \( DEF \), where \( A \) corresponds to \( D \) and \( C \) corresponds to \( F \). Angles \( C \) and \( F \) are right angles. If \( tan(A)=\surd 3 \) and \( DF=53 \), what is the length of \( DE \)?
原库参考解析
三角函数・单位圆与弧度挑战
In the xy-plane, a unit circle with center at the origin O contains point A with coordinates (1, 0) and point B with coordinates\( \left(-\frac{5}{\sqrt{89}},\frac{8}{\sqrt{89}}\right) \). If the measure of angle AOB is w radians, what is the value of \( tanw \)?
原库参考解析
三角函数・单位圆与弧度进阶
In the right triangle shown, what is the value of \( tanA \) ?
原库参考解析
三角函数・单位圆与弧度进阶
For two acute angles, ∠Q and ∠R , \( cos(Q)=sin(R) \). The measures, in degrees, of ∠Q and ∠R are \( x+61 \) and \( 4x+4 \), respectively. What is the value of x?
原库参考解析
三角函数・单位圆与弧度进阶
\( RS=504 \)
\( ST=128 \)
\( TR=520 \)
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 \( tanW \)?
原库参考解析
