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 triangle shown, RS= \( \sqrt{133} \) . 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
In ΔABC, ∠B is a right angle and the length of \( BC \) is 260 millimeters. If cos A = \( \frac{3}{5} \), what is the length. in millimeters, of \( AB \) ?
原库参考解析
三角函数・单位圆与弧度进阶
What is the value of cos A in the triangle shown?
原库参考解析
三角函数・单位圆与弧度进阶
Which of the following is equivalent to cos \( \frac{π}{3} \) ?
原库参考解析
三角函数・单位圆与弧度挑战
An angle has a measure of \( \frac{13π}{6} \) 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
In triangle XYZ, angle Y is a right angle, the measure of angle Z is 39 ° , and the length of \( YZ \) is 22 units. If the area, in square units, of triangle XYZ can be represented by the expression k tan 39 ° , 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
Point N lies on a unit circle in the xy-plane and has coordinates (1,0). Point O is the center of the circle and has coordinates (0,0). Point M also lies on the unit circle, and the measure of angle NOM is \( \frac{272π}{3} \) radians. If the coordinates of point M are (a, b), where a and b are constants, what is the value of a ?
原库参考解析
三角函数・单位圆与弧度进阶
RS = 320
ST =156
TR = 356
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・单位圆与弧度挑战
In triangle XYZ, angle Y is a right angle, the measure of angle Z is 31 °, and the length of \( YZ \) is 24 units. If the area, in square units, of triangle XYZ can be represented by the expression k tan 31°, 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 right triangle ABC, angles A and B are acute, side AC has a length of 29.5, and tanB = \( \frac{1}{9} \) . 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
Point F lies on a unit circle in the xy-plane and has coordinates (1, 0). Point G is the center of the circle and has coordinates (0, 0), Point H also lies on the circle and has coordinates (-1, y), where y is a constant. Which of the following could be the positive measure of angle FGH, in radians?
原库参考解析
三角函数・单位圆与弧度进阶
In right triangle ABC, angles A and B are acute, side AC has a length of 23.2, and\( Tan\ B=\frac{1}{7} \) . What is the length of side BC, rounded to the nearest tenth?
原库参考解析
