 UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

### Theory:

We will derive the trigonometric ratios of $$90^{\circ}$$ with the help of a unit circle.

A circle with radius $$1$$ unit centred at the origin is called a unit circle. Here $$OQ$$ $$=$$ $$OP$$ $$=$$ $$OC$$ $$=$$ $$1$$ unit (Radius).

Let us consider the first quadrant. Let the point $$C(x,y)$$ be any point on the unit circle and $$\angle COB$$ $$=$$ $$\theta$$.

In the right angles triangle $$COB$$ we have:

Opposite side $$=$$ $$y$$

Adjacent side $$=$$ $$x$$

Hypotenuse $$=$$ $$1$$

Now, let us determine the trigonometric ratios in the first quadrant with the coordinate $$C$$.

• Sine $$\theta$$:

$$\sin \theta$$ $$=$$ $$\frac{\text{Opposite side}}{\text{Hypotenuse}}$$

$$=$$ $$\frac{y}{1}$$

$$=$$ $$y$$

• Cosine $$\theta$$:

$$\cos \theta$$ $$=$$ $$\frac{\text{Adjacent side}}{\text{Hypotenuse}}$$

$$=$$ $$\frac{x}{1}$$

$$=$$ $$x$$

• Tangent $$\theta$$:

$$\tan \theta$$ $$=$$ $$\frac{\text{Opposite side}}{\text{Adjacent side}}$$

$$=$$ $$\frac{y}{x}$$

When $$\theta = 90^{\circ}$$, $$OC$$ coincides with $$OQ$$ then $$Q = (0,1)$$ where $$x$$ $$=$$ $$0$$ and $$y$$ $$=$$ $$1$$.

Then the trigonometric ratios are given by:

• Sine $$90^{\circ}$$:

$$\sin 0^{\circ}$$ $$=$$ $$y$$

$$=$$ $$1$$

• Cosine $$90^{\circ}$$:

$$\cos 90^{\circ}$$ $$=$$ $$x$$

$$=$$ $$0$$

• Tangent $$90^{\circ}$$:

$$\tan 90^{\circ}$$ $$=$$ $$\frac{1}{0}$$

$$=$$ 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 $$0$$