1092575吳姿瑩

三角函數，是人們用來表三角形上邊長與邊長之間關係的函數，當我們觀察一個直角三角形時，我們可以將各個函數定義作如下：

正弦$\sin( \theta )=\frac{對邊}{斜邊}$，餘弦$\cos( \theta )=\frac{臨邊}{斜邊}$

正切$\tan( \theta )=\frac{對邊}{臨邊}$，餘切$\cot( \theta )=\frac{臨邊}{對邊}$

正割$\sec( \theta )=\frac{斜邊}{臨邊}$，餘割$\csc( \theta )=\frac{斜邊}{對邊}$

$$\tan( \theta )=\frac{\sin( \theta )}{\cos( \theta )}，\cot( \theta )=\frac{\cos( \theta )}{\sin( \theta )}$$

$$\sec( \theta )=\frac{1}{\cos( \theta )}，\csc( \theta )=\frac{1}{\sin( \theta )}$$

$$\sin^{2}( \theta )\cdot\csc^{2}( \theta )=1$$

$$\cos^{2}( \theta )\cdot\sec^{2}( \theta )=1$$

$$\tan^{2}( \theta )\cdot\csc^{2}( \theta )=1$$

$$\sin^{2}( \theta )+\cos^{2}( \theta )=1$$

$$1+\tan^{2}( \theta )=\sec^{2}( \theta )$$

$$1+\cot^{2}( \theta )=\csc^{2}( \theta )$$

$$\sin(90^{\circ}-\theta)=\cos(\theta)$$

$$\cos(90^{\circ}-\theta)=\sin(\theta)$$

$$\sin\big(\alpha+\beta\big)=\sin\alpha\cdot\cos\beta+\cos\alpha\cdot\sin\beta$$

$$\sin\big(\alpha-\beta\big)=\sin\alpha\cdot\cos\beta-\cos\alpha\cdot\sin\beta$$

$$\cos\big(\alpha+\beta\big)=\cos\alpha\cdot\cos\beta-\sin\alpha\cdot\sin\beta$$

$$\cos\big(\alpha-\beta\big)=\cos\alpha\cdot\cos\beta+\sin\alpha\cdot\sin\beta$$

$$\tan\big(\alpha+\beta\big)= \frac{\tan \alpha +\tan \beta }{1-\tan \alpha \cdot\tan \beta }$$

$$\tan\big(\alpha-\beta\big)= \frac{\tan \alpha -\tan \beta }{1+\tan \alpha \cdot\tan \beta }$$

$$\cos(2\theta)=\cos^{2}\theta-\sin^{2}\theta$$

$$\tan(2\theta)= \frac{2\cdot\tan\theta}{1-\tan^{2}\theta}$$

$$\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1-\cos\theta}{2}}$$

$$\cos(\frac{\theta}{2})=\pm\sqrt{\frac{1+\cos\theta}{2}}$$

$$\tan(\frac{\theta}{2})=\pm\frac{\sin\theta}{1+\cos\theta}$$