PUMPA - THE SMART LEARNING APP

Helps you to prepare for any school test or exam

Download now on Google Play
Based 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 \(\theta\).
 
intro.png
 
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
sin.png
cos.png
tan.png
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, \(\cot \theta \cdot \tan \theta = 1\).