Trigonometric Functions
Term | Definition | Formula | Domain | Range | |||||
|---|---|---|---|---|---|---|---|---|---|
| Sine | In a right triangle, the ratio of the length of the side opposite the angle to the length of the hypotenuse. | \( \sin(\theta) = \frac{{\text{Opposite}}}{{\text{Hypotenuse}}} \) | \( (-\infty, \infty)\) | \([-1, 1]\) | |||||
| Cosine | In a right triangle, the ratio of the length of the adjacent side to the length of the hypotenuse | \( \cos(\theta) = \frac{{\text{Adjacent}}}{{\text{Hypotenuse}}} \) | \( (-\infty, \infty)\) | \([-1, 1]\) | |||||
| Tangent | In a right triangle, the ratio of the length of the side opposite the angle to the length of the adjacent side. | \( \tan(\theta) = \frac{{\text{Opposite}}}{{\text{Adjacent}}} \) | \(\frac{\pi}{2} + \pi n \text{, } n \text{ is an integer}\) | \( (-\infty, \infty)\) | |||||
| Secant | The reciprocal of the cosine of an angle. | \( \sec(\theta) = \frac{1}{{\cos(\theta)}} \) | \(\frac{\pi}{2} + \pi n \text{, } n \text{ is an integer}\) | \([-1, 1]\) | |||||
| Cotangent | The reciprocal of the tangent of an angle. | \( \cot(\theta) = \frac{1}{{\tan(\theta)}} \) | \(\pi n \text{, } n \text{ is an integer} \) | \( (-\infty, \infty)\) | |||||
| Cosecant | The reciprocal of the sine of an angle. | \( \csc(\theta) = \frac{1}{{\sin(\theta)}} \) | \(\pi n \text{, } n \text{ is an integer}\) | \([-1, 1]\) | |||||
| Radian | A unit of angular measure defined as the angle subtended by an arc of a circle that has the same length as the radius of the circle. One radian is equal to approximately 57.29 degrees. | \(\pi \) | |||||||
Inverse Trigonometric Functions
Term | Definition | Domain | Range | ||||
|---|---|---|---|---|---|---|---|
\(\arcsin(x)\) \(\sin^{-1}(x) \) | The inverse of the sine function; it returns the value \(y\) of the sine of \(x\). | \([-1,1]\) | \([-\frac{\pi}{2}, \frac{\pi}{2}]\) | ||||
\(\arccos(x)\) \(\cos^{-1}(x)\) | The inverse of the cos function; it returns the value \(y\) of the cos of \(x\). | \([-1,1]\) | \([0, \pi]\) | ||||
\(\arctan(x)\) \(\tan^{-1}(x)\) | The inverse of the tangent function; it returns the value \(y\) of the tan of \(x\). | \( (-\infty, \infty)\) | \([-\frac{\pi}{2}, \frac{\pi}{2}]\) | ||||
\(\sec^{-1}(x)\) | The inverse of the sec function; it returns the value \(y\) of the sec of \(x\). | \((-\infty, -1)\cup(1, \infty)\) | \([0, \pi], y \neq \frac{\pi}{2} \) | ||||
\(\csc^{-1}(x)\) | The inverse of the csc function; it returns the value \(y\) of the csc of \(x\). | \((-\infty, -1]\cup[1, \infty)\) | \([-\frac{\pi}{2}, \frac{\pi}{2}], y \neq 0\) | ||||
\(\cot^{-1}(x)\) | The inverse of the cot function; it returns the value \(y\) of the cot of \(x\). | \( (-\infty, \infty)\) | \([0, \pi]\) | ||||
Special Values for Trigonometric Functions
Function | 0° | 30° | 45° | 60° | 90° | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| sin | \(0\) | \(\frac{1}{2} \) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) | ||||||
| cos | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2} \) | \(0\) | ||||||
| tan | \(0\) | \(\frac{\sqrt{3}}{3}\) | \(1\) | \(\sqrt{3}\) | Not defined | ||||||
| cot | Not defined | \(\sqrt{3}\) | \(1\) | \(\frac{\sqrt{3}}{3}\) | \(0\) | ||||||
| csc | Not defined | \(2\) | \(\sqrt{2}\) | \(\frac{2\sqrt{3}}{3}\) | \(1\) | ||||||
| sec | \(1\) | \(\frac{2\sqrt{3}}{3}\) | \(\sqrt{2}\) | \(2\) | Not defined | ||||||
Trigonometric Identities
Term | Definition | ||
|---|---|---|---|
| Pythagorean Identities |
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| Sum and Difference Identities |
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| Double Angle Identities |
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| Half Angle Identities |
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| Product to Sum Identities |
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| Sum to Product Identities |
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| Law of Sines | \(\frac{{a}}{{\sin A}} = \frac{{b}}{{\sin B}} = \frac{{c}}{{\sin C}}\) | ||
| Law of Cosines |
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Graphs of Trigonometric Functions
Term | Definition | ||
|---|---|---|---|
| Amplitude | One half of the difference between a graph's highest value and lowest value. | ||
| Period | The interval it takes for a graph to complete one cycle. | ||
| Cycle | One complete movement between a starting point, a graph's highest point, its lowest point, and finally, its starting point again. | ||
| Frequency | The amount of cycles that happen in one second. | ||
Horizontal Shift Phase Shift | The graph's offset on the horizontal axis. | ||
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