Theory:

We will derive the trigonometric ratios of \(90^{\circ}\) using the right angled triangle \(ABC\).
 
90_deg.png
 
Let us now experiment with the given triangle concerning \(\angle A\).
 
Increase the value of \(\theta\) to the extent it becomes \(90\) degree.
 
90_deg.gif
 
It is observed that as \(\angle A\) gets larger and larger, the point \(A\) gets closer to the point \(B\).
 
That is, when \(\theta\) becomes very close to \(90^{\circ}\), the side \(AC\) becomes at most the same as the side \(BC\).
 
This implies that the measure of \(AB\) becomes almost zero.
 
In the right angles triangle \(ABC\) we have:
 
Opposite side \(=\) \(BC\)
 
Adjacent side \(=\) \(AB\)
 
Hypotenuse \(=\) \(AC\)
 
Now, let us determine the trigonometric ratios when \(\theta = 90^{\circ}\) as follows.
 
  • Sine \(90^{\circ}\):
 
\(\sin \theta\) \(=\) \(\frac{\text{Opposite side}}{\text{Hypotenuse}}\)
 
\(\sin \theta\) \(=\) \(\frac{BC}{AC}\)
 
\(\sin 90^{\circ}\) \(=\) \(1\) [When \(\angle A = 90^{\circ}\), \(AC\) \(=\) \(BC\).]
 
  • Cosine \(90^{\circ}\):
 
\(\cos \theta\) \(=\) \(\frac{\text{Adjacent side}}{\text{Hypotenuse}}\)
 
\(\cos \theta\) \(=\) \(\frac{AB}{AC}\)
 
\(\cos 90^{\circ}\)\(=\) \(\frac{0}{AC}\)
 
\(=\) \(0\)
 
  • Tangent \(90^{\circ}\):
 
\(\tan 90^{\circ}\) \(=\) \(\frac{\sin 90^{\circ}}{\cos 90^{\circ}}\)
 
\(=\) \(\frac{1}{0}\)
 
\(=\) not defined
 
Using these basic trigonometric ratios determine their reciprocals as follows:
 
  • Cosecant \(90^{\circ}\):
 
\(\text{cosec}\,90^{\circ}\) \(=\) \(\frac{1}{\sin 90^{\circ}}\)
 
\(=\) \(\frac{1}{1}\)
 
\(=\) \(1\)
 
  • Secant \(90^{\circ}\):
 
\(\sec 90^{\circ}\) \(=\) \(\frac{1}{\cos 90^{\circ}}\)
 
\(=\) \(\frac{1}{0}\)
 
\(=\) not defined
 
  • Cotangent \(90^{\circ}\):
 
\(\cot 90^{\circ}\) \(=\) \(\frac{1}{\tan 90^{\circ}}\)
 
\(=\) \(\frac{0}{1}\)
 
\(=\) \(0\)
 
Let us summarize all the trigonometric ratios of \(90^{\circ}\) in the following table.
 
 
\(\sin \theta\)
\(\cos \theta\)
\(\tan \theta\)
\(\text{cosec}\,\theta\)
\(\sec \theta\)
\(\cot \theta\)
\(\theta = 90^{\circ}\)
\(1\)
\(0\)
not defined
\(1\)
not defined
\(0\)