- Definition Let $P(x,y)$ be a point on the rectangular coordinate plane, where $O$ is the origin. The line segment $OP$ is the terminal side of the angle of rotation $\theta$ and the length of $OP$ is $r$. Note that $r=\sqrt{x^2+y^2}$. The six trigonometric functions are as follow:
- $\sin\theta = \dfrac{y}{r}$
- $\cos\theta = \dfrac{x}{r}$
- $\tan\theta = \dfrac{y}{x}$
- $\sec\theta = \dfrac{1}{\cos\theta} = \dfrac{r}{x}$
- $\csc\theta = \dfrac{1}{\sin\theta} = \dfrac{r}{y}$
- $\tan\theta = \dfrac{1}{\tan\theta} = \dfrac{x}{y}$
- The Trigonometric Identities
- Reciprocal Identities
- $\csc\theta=\dfrac{1}{\sin\theta}$
- $\sec\theta=\dfrac{1}{\cos\theta}$
- $\cot\theta=\dfrac{1}{\tan\theta}$
- Quotient Identities
- $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$
- $\cot\theta=\dfrac{\cos\theta}{\sin\theta}$
- Square Identities
- $\sin^2\theta+\cos^2\theta=1$
- $1+\tan^2\theta=\sec^2\theta$
- $1+\cot^2\theta=\csc^2\theta$
- Identities involving $\dfrac{n\pi}{2}\pm\theta$
$$
\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline
& -\theta & \dfrac{\pi}{2}-\theta & \dfrac{\pi}{2}+\theta & \pi-\theta & \pi+\theta & \dfrac{3\pi}{2}-\theta & \dfrac{3\pi}{2}+\theta & 2\pi-\theta \\ \hline
\sin & -\sin\theta & \cos \theta & \cos\theta & \sin\theta & -\sin\theta & -\cos\theta & -\cos\theta & -\sin\theta \\ \hline
\csc & -\csc\theta & \sec\theta & \sec\theta & \csc\theta & -\csc\theta & -\sec\theta & -\sec\theta & -\csc\theta \\ \hline
\cos & \cos\theta & \sin\theta & -\sin\theta & -\cos\theta & -\cos\theta & -\sin\theta & \sin\theta & \cos\theta \\ \hline
\sec & \sec\theta & \csc\theta & -\csc\theta & -\sec\theta & -\sec\theta & -\csc\theta & \csc\theta & \sec\theta \\ \hline
\tan & -\tan\theta & \cot\theta & -\cot\theta & -\tan\theta & \tan\theta & \cot\theta & -\cot\theta & -\tan\theta \\ \hline
\cot & -\cot\theta & \tan\theta & -\tan\theta & -\cot\theta & \cot\theta & \tan\theta & -\tan\theta & -\cot\theta \\ \hline
\end{array}
$$
- Reciprocal Identities
Resources for HKDSE Mathematics
