PDF chapter test TRY NOW
We will derive the trigonometric ratios of \(90^{\circ}\) using the right angled triangle \(ABC\).
![90_deg.png](https://resources.cdn.yaclass.in/d85177e0-f298-4d1d-b071-7e4684706638/90degw200.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](https://resources.cdn.yaclass.in/62e0c238-5bf9-43a6-92bf-41f654275a8f/90deg.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 | \(0\) |