**Trigonometric Ratios Table**

__Trigonometric Ratios of some specific angles and its Derivation__

** 1. Derivation of the value of sin 0****° **and **cos 0****°**

** ** Let us consider a Right-angled triangle, △ABC right angled at A that is ∠BAC = 90°

We know that,

Now, if the value of angle **θ **approaches to 0**° or **become 0**°**, then in rt. △ABC, the length of Perpendicular AC will be decreased gradually, finally become **0** and accordingly, Hypotenuse BC coincides or overlap with Base AB (BC = AB)

** 2 . Derivation of the value of sin 90****° **and **cos 90****°**

Let us consider a Right-angled triangle, △ABC right angled at A that is ∠BAC = 90°

We know that,

Now, if the value of angle **θ **approaches to 90**° or **become 90**°**, then in rt. △ABC, the length of Base AB will be decreased gradually, finally become **0** and accordingly, Hypotenuse BC coincides or overlap with Base AC (BC = AC)

** 3. Derivation of the value of sin 45****°and cos 45****°:**

Let us consider a Right-angled triangle, △PQR right angled at Q that is ∠PQR = 90°

In △PQR, we have

Again, according to Pythagoras Theorem, we have

Similarly,

** 4. Derivation of the value of sin 30****°, sin 60****°and cos 30****°, cos 60****°:**

Let us consider a Equilateral Triangle, △ABC such that AB = BC = AC = a (say)

Now, in rt. △ABD, we have

Applying Pythagoras Theorem in rt. △ABD, we get

and so, we have in rt. △ABD,

**Preparation of Trigonometric Table for Trigonometric Ratios of specific angles**

In order to find out the value of Trigonometric Ratios of specific angles, perform the following calculation mentioned as under:

- Fill all the column along
**sin θ **in the series as 0, 1, 2, 3 and 4
- Fill all the column along
**cos θ **the series as 4, 3, 2, 1 and 0
- Divide all the entries by 4 and calculate the square root of it

In simplified form, the value of **Trigonometric Ratios of specific angles **will be obtained as

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