My name is Han Yin, and I have developed this website as a resource to facilitate the review of key concepts and it is not fully completed. Feel free to email me at hanyin@ku.edu if there are any errors or suggestions for improvement.
\[ \sin(\theta) = \frac{1}{\csc(\theta)}, \qquad \cos(\theta) = \frac{1}{\sec(\theta)}, \qquad \tan(\theta) = \frac{1}{\cot(\theta)}, \qquad \csc(\theta) = \frac{1}{\sin(\theta)}, \qquad \sec(\theta) = \frac{1}{\cos(\theta)}, \qquad \cot(\theta) = \frac{1}{\tan(\theta)}. \]
\[ \sin^2(\theta) + \cos^2(\theta) = 1, \quad 1 + \tan^2(\theta) = \sec^2(\theta), \quad 1 + \cot^2(\theta) = \csc^2(\theta). \]
\[ \sin(-\theta) = -\sin(\theta), \quad \cos(-\theta) = \cos(\theta), \quad \tan(-\theta) = -\tan(\theta), \quad \csc(-\theta) = -\csc(\theta), \quad \sec(-\theta) = \sec(\theta), \quad \cot(-\theta) = -\cot(\theta). \]
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
\[ \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}, \quad \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}. \]
The unit circle is a circle with a radius of 1. It is used to define the trigonometric functions of any angle.
| Function | Midline | Amplitude | Vertical Asymptote | Period | Phase Shift |
|---|---|---|---|---|---|
| \( A \cos(Bx - C) + D \) | \( y = D \) | \( |A| \) | No | \( \frac{2\pi}{|B|} \) | \( \frac{C}{B} \) |
| \( A \sin(Bx - C) + D \) | \( y = D \) | \( |A| \) | No | \( \frac{2\pi}{|B|} \) | \( \frac{C}{B} \) |
| \( A \tan(Bx - C) + D \) | \( y = D \) | \(|A|\) | For \( \tan \): \( \frac{\pi}{2B} + \frac{C}{B} + \frac{\pi k}{B} \) | \( \frac{\pi}{|B|} \) | \( \frac{C}{B} \) |
| \( A \cot(Bx - C) + D \) | \( y = D \) | \(|A|\) | \( \frac{\pi}{B} + \frac{C}{B} + \frac{\pi k}{B} \) | \( \frac{\pi}{|B|} \) | \( \frac{C}{B} \) |
| \( A \sec(Bx - C) + D \) | \( y = D \) | \( |A| \) | \( \frac{\pi}{2B} + \frac{C}{B} + \frac{\pi k}{B} \) | \( \frac{2\pi}{|B|} \) | \( \frac{C}{B} \) |
| \( A \csc(Bx - C) + D \) | \( y = D \) | \( |A| \) | \( \frac{\pi}{B} + \frac{C}{B} + \frac{\pi k}{B} \) | \( \frac{2\pi}{|B|} \) | \( \frac{C}{B} \) |
| Function | Domain | Range |
|---|---|---|
| \( \arcsin(x) \) | \( -1 \leq x \leq 1 \) | \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \) |
| \( \arccos(x) \) | \( -1 \leq x \leq 1 \) | \( 0 \leq y \leq \pi \) |
| \( \arctan(x) \) | \( -\infty \lt x \lt \infty \) | \( -\frac{\pi}{2} \lt y \lt \frac{\pi}{2} \) |
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]
The length of the Transverse axis is \(2a\).
The coordinates of the vertices are \((\pm a, 0)\).
The length of the conjugate axis is \(2b\).
The coordinates of the co-vertices are \((0, \pm b)\).
The distance between of the foci are \((\pm c, 0)\), where \(c = \sqrt{a^2 + b^2}\).
The equations of the asymptote are \(y = \pm \frac{b}{a}x\).
\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]
The length of the Transverse axis is \(2a\).
The coordinates of the vertices are \((0, \pm a)\).
The length of the conjugate axis is \(2b\).
The coordinates of the co-vertices are \((\pm b, 0)\).
The coordinates of the foci are \((0, \pm c)\), where \(c = \sqrt{a^2 + b^2}\).
The distance between of the foci are \(2c\).
The equations of the asymptote are \(y = \pm \frac{a}{b}x\).
\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]
\[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]
\[y^2 = 4px\]
\[x^2 = 4py\]