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
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
Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles C and F are right angles. The length of \( AB \) is 2.8 times the length of \( DE \) . If \( tanA=\frac{20}{21} \) , what is the value of sin D?
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
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
The measure of angle F is \( \frac{7π}{4} \) radians. What is the value of cos(F) ?
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, cosQ = sinR. The measures, in degrees, of ∠Q and ∠R are x + 27 and 4x ＋ 3, 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
The measure of angle D is \( \frac{5π}{4} \) radians. What is the value of sin(D)?
原库参考解析
三角函数・单位圆与弧度进阶
ln triangle ABC, the measure of angle C is 90 degrees. The cosine of angle A is \( \frac{60}{61} \) . what is the sine of angle B?
原库参考解析
三角函数・单位圆与弧度挑战
Right triangles ABC and DEF shown above are similar, where A corresponds to D and B corresponds to E. If tan(A) = \( \frac{15}{25} \), what is tan(F) ?
原库参考解析
三角函数・单位圆与弧度进阶
Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles C and F are right angle. The length of AB is 2.4 times the length of DE. If tan A=21/20, what is the value of sin D?
原库参考解析
三角函数・单位圆与弧度进阶
Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angle C and F are right angles. The length of AB is 2.4 times the length of DE. If tanA= \( \frac{21}{20} \) , what is the value of sinD ?
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 sin y "?
原库参考解析
