Trigonometry is a branch of mathematics which deals with the study of triangles,ratios of triangles,etc.In class 10,we will study about trigonometric ratios, values of some trigonometric ratios, and some properties of trigonometric functions.

Let’s go

back to the time where we are pirates of the sea searching for gold.

We don’t

have any gps here, so how can we find our position and location.

**How?**

The answer

is **trigonometry.**

Trigonometry is made from two greek words , **trigon**, meaning triangles and **metron**, meaning measure.

So,

trigonometry is the study of angles, ratios and sides of triangles.

We have a

triangle, we have three sides, three angles and that’s it.

Let’s get a feel of trigo by getting into a situation.

## trigonometry -> the situation

Suppose we

are on a boat which is a few metres away

from a light house.

Now , what

if we want to find the distance of the boat from the light house ?

It is assumed

that the height of the light house is

known .

What if we also

know the angle which light house makes from our boat ?

We can

easily find the distance.

**HOW ?**

Do you know that the ratio of two sides of a right angled triangle is fixed if one of the angle is fixed ?

So, p/b = c

(a fixed no.)

From the

above equation, we can easily find b, as p and c are known.

Isn’t it

amazing ?

Now, as we are humans we can’t believe anything

without proof.

So, we must

have a proof of this statement :

**ratio of any two sides of a right angled triangle is fixed if one of the angle is fixed.**

## trigonometry->the proof

Suppose we

have a triangle ABC, where

AB = 2cm, AC = 4cm,

BC = √12 cm.

Now, angle

ACB = 30°.

Here, AB/AC

= 1/2.

If we just

double the length of AB and fix the angle to be 30°.

Then, we

have to increase the lengths of AC and BC.

Now, after

measuring AC, we have AB = 4cm and AC = 8 cm.

Here also,

AB/AC = 4/8 = 1/2.

Hence,

proved that if the angle is fixed , then the ratios of the lengths of any two

sides is a constant.

Now, for different angles, we have different constants.

**trigonometry-> ratios**

Also, we

have six different ratios.

**HOW ?**

Just see

that we have three sides of a triangle,

Here, these

are : hypotenuse( the longest side) ,H

Perpendicular,

P

Base ,B

Now, we have

the following ratios :

P/H B/H

P/B

H/P H/B

B/P

Now for different angles we have different value of P/H .

**trigonometry-> the naming**

So, we can’t

just name it as anything.

We have to

name it in such a way that the name contains the angle so that no one get

confuse.

We name it

as sine .

Now, from

the above example, sine of 30 degree is 1/2.

Now it is

very boring and time taking to write sine of 30 , so we denote it as sin(30) =

1/2.

Similarly,

for every ratio, we have a function , called trigonometric function which takes

angle as input and give the output as a constant.

Following

are the names, formulae and abbreviations of these functions.

P/H sine sin

B/H cosine cos

P/B

tangent tan

H/P cosecant cosec

H/B secant sec

B/P cotangent cot

## trigonometry->the properties

What we have

observed ?

cosecant is

inverse of sine.

secant is

inverse of cosine.

cotangent is

inverse of tangent.

Also , sin/cos = p/h / b/h = p/b = tan

So, tan(x)=

sin(x)/ cos(x).

Similarly,

cot(x)= cos(x)/sin(x).

Now, we will

study some properties of these functions :

sin^{2}x

+ cos^{2}x = 1.

tan^{2}x

+ 1 = sec^{2}x.

1 + cot^{2}x

= cosec^{2}x.

To know the proof of them, just see the following post :

Here are solutions of trigonometry class 10 NCERT