Introduction to Trigonometry .

Class X

Subject: Mathematics 

The ratios of sides in a right angle triangle is said to be trigonometric ratio.
The ratios of sides are as follows.
1. Perpendicular to hypotenuse
2. Base to hypotenuse
3. Perpendicular to base
4. Base to perpendicular
5. Hypotenuse to base
6. Hypotenuse to perpendicular

These ratios of sides are generally denoted by
1. sine (sin)
2. cosine (cos)
3. tangent (tan)
4. cotangent (cot)
5. secant (sec)
6. cosecant (cosec)

1.The opposite side of considering angle is identified as perpendicular.
2. The adjacent side of considering angle is identified as base.
3. The opposite side of right angle is identified as hypotenuse.

The right triangle ABC in which B is right angle. The considering angle can be either A or C.
When considering angle is C.
1.Perpendicular will be AB.
2. Base will be BC.
3. Hypotenuse will be AC.

When considering angle is A.
1. Perpendicular will be BC.
2. Base will be AB.
3. Hypotenuse will be AC.

There will be no change in the value of hypotenuse in both cases.

When considering angle is A.
The value of trigonometric ratios will be as follows.
1. sinA = BC/AC
2. cosA = AB/AC
3. tanA = BC/AB
4. cotA = AB/BC
5. secA = AC/AB
6. cosecA = AC/BC

When considering angle is C.
The value of trigonometric ratios will be as follows.

1. sinC = AB/AC
2. cosC = BC/AC
3. tanC = AB/BC
4. cotC = BC/AB
5. secA = AC/BC
6. cosecA = AC/AB

Here, the sum of angles A and C will be complementary or right angle.
Therefore,
A = 90⁰ - C
C = 90⁰ - A.
So,
sinA = cosC
cosA = sinC
tanA = cotC
cotA = tanC
secA = cosecC
cosecA = secC

We consider some special trigonometric ratios of 0⁰,30⁰, 45⁰, 60⁰ and 90⁰.

1.For 45⁰, we consider in an  isosceles triangle.
2. For 30⁰ and 60⁰ consider in an equilateral triangle.
3. For 0⁰ and 90⁰ coinsides the perpendicular and base with hypotenuse respectively.

All relations which are derived by Pythagoras theorem is said to be trigonometrical identities which is true for all values of considering angles.

In Right Angle Triangle ABC, in which angle B is right angle.

If considering angle is C which is θ.

AB² + BC² = AC²
When both sides divided by AC².

1. sin²θ + cos²θ = 1

This is a trigonometrical identity and true for all values of θ.

In this way, we find out rest other trigonometrical identities by dividing both sides by BC² and AC² respectively.

Rest others are as follows
4. 1 + tan²θ = sec²θ
5. sec²θ - tan²θ = 1
6. sec²θ - 1 = tan²θ
7. 1 + cot²θ = cosec²θ
8. cosec²θ - cot²θ = 1
9. cosec²θ - 1 = cot²θ

Others trigonometrical identities are derived from reciprocal of ratio of sides and complementary angle of rest angles of triangle.

As per reciprocal,
1. sinθ.cosecθ = 1
2. cosθ.secθ = 1
3. tanθ.cotθ = 1

As per complementary angle.

1. sinθ = cos(90⁰ - θ)
2. cosθ = sin(90 - θ)
3. tanθ = cot(90⁰ - θ)
4. cotθ = tan(90⁰ - θ)
5. secθ = cosec(90⁰ - θ)
6. cosecθ = sec(90⁰ - θ)

These are trigonometrical identities and true for each values of considering angle in right angle triangle.