### Theory:

We will derive the trigonometric ratios of $$90^{\circ}$$ using the right angled triangle $$ABC$$.

Let us now experiment with the given triangle concerning $$\angle A$$.

Increase the value of $$\theta$$ to the extent it becomes $$90$$ degree.

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 $$0$$