# Trigonometric Ratios or Functions

In this section we will discuss about trigonometric ratios which are also know as trigonometric functions and later we will discuss identities involving them.Consider an angle $\Theta$ = $\angle$ BOA. Let P(x,y) be any point other than O on its terminal side AO. Draw a perpendicular from P on ray OB. Let OA = r, OM = x and PM = y. OP which is r is always positive while x and y can be positive or negative. It depends on the position of the terminal side OA of $\angle$ BOA.

In right angled triangle OMP, we have

Base = OM = x and perpendicular = PM = y and hypotenuse OP = r

On the basis of this we have six trigonometric ratios.

1) $sin \Theta = \frac{Perpendicular}{Hypotenuse} = \frac{y}{r}$

2) $cos \Theta = \frac{Base}{Hypotenuse} = \frac{x}{r}$

3) $tan \Theta = \frac{Perpendicular}{Base} = \frac{y}{x}$

4) $csc \Theta = \frac{Hypotenuse}{Perpendicular} = \frac{r}{y}$

5) $sec \Theta = \frac{Hypotenuse}{Base} = \frac{r}{x}$

6) $cot \Theta = \frac{Base}{Perpendicular} = \frac{x}{y}$

The triangle OMP is known as the triangle of reference.

**Note :**$sin \Theta$ does not mean the product of sin and $ \Theta$. The $sin \Theta$ is read as 'sine of angle $ \Theta $.

These functions depend only on the value of the angle $ \Theta$ and not on the position of the point 'P' chosen on the terminal side of the angle $ \Theta$.

## Trigonometric ratios are same for the same angle.

Let point P(x,y) and point Q(x', y') be two different points on the terminal side OA with OP as r and OQ as r'. So there are two right triangles, triangle POM and triangle QON. These two triangles are similar.∴ $\frac{y}{r} = \frac{y'}{r'}$, $\frac{x}{r} = \frac{x'}{r'}$ and $\frac{y}{x} = \frac{y'}{x'}$

⇒ $sin \Theta = \frac{y}{r}$ ⇒ $sin \Theta = \frac{y'}{r'}$

So, $sin \Theta = \frac{y}{r} = \frac{y'}{r'}$

⇒ $cos \Theta = \frac{x}{r}$ ⇒ $sin \Theta = \frac{x'}{r'}$

So, $cos \Theta = \frac{x}{r} = \frac{x'}{r'}$

⇒ $tan \Theta = \frac{y}{x}$ ⇒ $sin \Theta = \frac{y'}{x'}$

So, $tan \Theta = \frac{y}{x} = \frac{y'}{x'}$

Thus, sine, cosine and tangent are the same whatever point to be taken on the terminal side OA. Similarly it is true for other ratios also.

**Remark 1 :**If the terminal side and x-axis are same then $csc \Theta$ and $cot \Theta$ are not defined. If the terminal side and y-axis are same then $sed \Theta$ and $tan \Theta$ are not defined.

**Remark 2 :**Relations between trigonometric ratios :

i) $sin \Theta \times csc\Theta = 1 \Rightarrow sin \Theta = \frac{1}{csc\Theta }$ and $csc \Theta = \frac{1}{sin\Theta}$

ii) $cos \Theta \times sec\Theta = 1 \Rightarrow cos \Theta = \frac{1}{sec\Theta }$ and $sec \Theta = \frac{1}{cos\Theta}$

iii) $tan \Theta \times cot\Theta = 1 \Rightarrow tan \Theta = \frac{1}{cot\Theta }$ and $cot \Theta = \frac{1}{tan\Theta}$

iv) $tan\Theta = \frac{sin\Theta }{cos\Theta }$ and $cot \Theta =\frac{cos\Theta }{sin\Theta}$

**Remark 3 :**Trigonometric ratios may be positive or negative, depends on the values of x and y.

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