Theory:

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, \(\cot \theta \cdot \tan \theta = 1\).
Important!
The ratios of tangent and cotangent are also given by the following quotients:
  • \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and
  • \(\cot \theta =  \frac{\cos \theta}{\sin \theta}\)