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.



11th grade math

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