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Book Free DemoBased on the three basic trigonometric ratios \(\sin\), \(\cos\) and \(\tan\) we will define its reciprocals.
Reciprocal Ratios:
Consider a right-angled triangle with a corresponding angle \(\theta\).
The three basic trigonometric ratios are:
- Sine
- Cosine
- Tangent
The table below depicts the relation of reciprocal ratios with the right-angled triangle.
| Name of the angle | Sine | Cosine | Tangent |
| Short form of the angle | \(\sin\) | \(\cos\) | \(\tan\) |
| Relationship | \(\sin \theta\) \(=\) \(\frac{\text{Opposite side}}{\text{Hypotenuse}}\) | \(\cos \theta\) \(=\) \(\frac{\text{Adjacent side}}{\text{Hypotenuse}}\) | \(\tan \theta\) \(=\) \(\frac{\text{Opposite side}}{\text{Adjacent side}}\) |
| Name of the reciprocal angle | Cosecant | Secant | Cotangent |
| Short form of the angle | \(\text{cosec}\) | \(\sec\) | \(\cot\) |
| Measurements related to the right-angled triangle |  |  |  |
| Relationship | \(\text{cosec}\,\theta\) \(=\) \(\frac{\text{Hypotenuse}}{\text{Opposite side}}\) | \(\sec \theta\) \(=\) \(\frac{\text{Hypotenuse}}{\text{Adjacent side}}\) | \(\cot \theta\) \(=\) \(\frac{\text{Adjacent side}}{\text{Opposite side}}\) |
| Relation with the basic ratio | \(\text{cosec}\,\theta\) \(=\) \(\frac{1}{\sin \theta}\) or \(\sin \theta\) \(=\) \(\frac{1}{\text{cosec}\,\theta}\) | \(\sec \theta\) \(=\) \(\frac{1}{\cos \theta}\) or \(\cos \theta\) \(=\) \(\frac{1}{\sec \theta}\) | \(\cot \theta\) \(=\) \(\frac{1}{\tan \theta}\) or \(\tan \theta\) \(=\) \(\frac{1}{\cot \theta}\) |
We can write certain identities based on these relationships.
- Identity \(1\):
\(\text{cosec}\,\theta \times \sin \theta\) \(=\) \(\text{cosec}\,\theta \times \frac{1}{\text{cosec}\,\theta}\)
\(=\) \(1\)
Therefore, \(\text{cosec}\,\theta \cdot \sin \theta= 1\).
- Identity \(2\):
\(\sec \theta \times \cos \theta\) \(=\) \(\sec \theta \times \frac{1}{\sec \theta}\)
\(=\) \(1\)
Therefore, \(\sec \theta \cdot \cos \theta = 1\).
- Identity \(3\):
\(\cot \theta \times \tan \theta\) \(=\) \(\cot \theta \times \frac{1}{\cot \theta}\)
\(=\) \(1\)
Therefore, \(\cot \theta \cdot \tan \theta = 1\).
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