Trigonometric ratios:
Let \(ABC\) be a right angled triangle.
Let \(0^\circ < \theta < 90^\circ\).
\(\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{AB}{AC}\)
\(\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{BC}{AC}\)
\(\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{AB}{BC}\)
\(\text{cosec}\: \theta = \frac{\text{Hypotenuse}}{\text{Opposite side}} = \frac{AC}{AB}\)
\(\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent side}} = \frac{AC}{BC}\)
\(\cot \theta = \frac{\text{Adjacent side}}{\text{Opposite side}} = \frac{BC}{AB}\)
\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
\(\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta}\)
\(\text{cosec}\: \theta = \frac{1}{\sin \theta}\)
\(\sec \theta = \frac{1}{\cos \theta}\)
Trigonometric ratio table:
Complementary angle:
| \(\sin (90^\circ - \theta) = \cos \theta\) | \(\cos (90^\circ - \theta) = \sin\theta\) |
| \(\tan (90^\circ - \theta) = \cot \theta\) | \(\cot (90^\circ - \theta) = \tan \theta\) |
| \(\text{cosec} (90^\circ - \theta) = \sec \theta\) |
\(\sec (90^\circ - \theta) = \text{cosec}\: \theta\)
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Trigonometric identities:
The fundamental identities of trigonometry are:
(1) \(\sin^2 \theta + \cos^2 \theta = 1\)
(2) \(1 + \tan^2 \theta = \sec^2 \theta\)
(3) \(1 + \cot^2 \theta = \text{cosec}^2\: \theta\)
The three trigonometric identities are true for every \(\theta\) lies between \(0^\circ\) and \(90^\circ\).
Equal forms of trigonometric identities:
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Identity
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Equal form of identity
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| \(\sin^2 \theta + \cos^2 \theta = 1\) | \(\sin^2 \theta = 1 - \cos^2 \theta \) (or) \(\cos^2 \theta = 1 - \sin^2 \theta\) |
| \(1 + \tan^2 \theta = \sec^2 \theta\) | \(\tan^2 \theta = \sec^2 \theta - 1\) (or) \(\sec^2 \theta - \tan^2 \theta = 1\) |
| \(1 + \cot^2 \theta = \text{cosec}^2\: \theta\) | \(\cot^2 \theta = \text{cosec}^2\: \theta - 1\) (or) \(\text{cosec}^2\: \theta - \cot^2 \theta = 1\) |
Important!
\((\sin \theta)^2 = \sin^2 \theta\)
\((\cos \theta)^2 = \cos^2 \theta\)
\((\tan \theta)^2 = \tan^2 \theta\)
\((\text{cosec}\: \theta)^2 = \text{cosec}^2\: \theta\)
\((\sec \theta)^2 = \sec^2 \theta\)
\((\cot \theta)^2 = \cot^2 \theta\)