UPSKILL MATH PLUS
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Learn moreBased on the three basic trigonometric ratios \(\sin\), \(\cos\) and \(\tan\) we will define its reciprocals.
Reciprocal Ratios:
Consider a rightangled 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 rightangled 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 rightangled 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, \(\cot \theta \cdot \tan \theta = 1\).
Important!
The ratios of tangent and cotangent are also given by the following quotients:
 \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and
 \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)