Trigonometric ratios
Trigonometric ratios in terms of sides
- \(\sin{\theta}=\frac{\text{Opposite Side}}{\text{Hypotenuse}}\)
- \(\cos{\theta}=\frac{\text{Adjacent Side}}{\text{Hypotenuse}}\)
- \(\tan{\theta}=\frac{\text{Opposite Side}}{\text{Adjacent Side}}\)
Trigonometric Radios
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Relation between trigonometric ratios
- \(\tan{\theta}=\frac{\sin{\theta}}{\cos{\theta}}\)
- \(\text{cosec}\,{\theta}=\frac{1}{\sin{\theta}}\)
- \(\sec{\theta}=\frac{1}{\cos{\theta}}\)
- \(\cot{\theta}=\frac{\cos{\theta}}{\sin{\theta}}=\frac{1}{\tan{\theta}}\)
Trigonometric identities
- \(\sin^2{\theta} + \cos^2{\theta} = 1\)
- \(\sec^2{\theta} - \tan^2{\theta} = 1\)
- \(\text{cosec}^2{\theta} - \cot^2{\theta} = 1\)
Trigonometric Identities
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Trigonometric ratios for common angles
| \(\theta\) | \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | \(90^\circ\) |
|---|---|---|---|---|---|
| \(\sin{\theta}\) | \(0\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(\cos{\theta}\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) | \(0\) |
| \(\tan{\theta}\) | \(0\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) | \(\text{undefined}\) |
| \(\text{cosec}\,\theta\) | \(\text{undefined}\) | \(2\) | \(\sqrt{2}\) | \(\frac{2}{\sqrt{3}}\) | \(1\) |
| \(\sec{\theta}\) | \(1\) | \(\frac{2}{\sqrt{3}}\) | \(\sqrt{2}\) | \(2\) | \(\text{undefined}\) |
| \(\cot{\theta}\) | \(\text{undefined}\) | \(\sqrt{3}\) | \(1\) | \(\frac{1}{\sqrt{3}}\) | \(0\) |
This table summarizes the values of sine, cosine, and tangent for common angles, expressed in degrees.
Trigonometric Radios for Some Standard Angles
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Signs in different quadrants
| Quadrant I | Quadrant II | Quadrant III | Quadrant IV | |
|---|---|---|---|---|
| \(\sin{\theta}\) | \(+\) | \(+\) | \(-\) | \(-\) |
| \(\cos{\theta}\) | \(+\) | \(-\) | \(-\) | \(+\) |
| \(\tan{\theta}\) | \(+\) | \(-\) | \(+\) | \(-\) |
Signs of Trigonometric Ratios
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Domains and ranges
| Function | Domain | Range |
|---|---|---|
| \(\sin{\theta}\) | \(R\) | \([-1, 1]\) |
| \(\cos{\theta}\) | \(R\) | \([-1, 1]\) |
| \(\tan{\theta}\) | \(R-\{(2n+1)\frac{\pi}{2}: n\in Z\}\) | \(R\) |
| \(\text{cosec}\,{\theta}\) | \(R-\{n\pi: n\in Z\}\) | \((-\infty,-1]\cup[1,\infty)\) |
| \(\sec{\theta}\) | \(R-\{(2n+1)\frac{\pi}{2}: n\in Z\}\) | \((-\infty,-1]\cup[1,\infty)\) |
| \(\cot{\theta}\) | \(R-\{n\pi: n\in Z\}\) | \(R\) |
These tables provide a summary of the values and properties of common trigonometric ratios for various angles, signs of ratios in different quadrants, and the domains and ranges of trigonometric functions.
Domain & Range of Trigonometric Ratios
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