# Trigonometric Ratios of some specific angles and its Derivation

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:

1. Fill all the column along sin θ in the series as 0, 1, 2, 3 and 4
2. Fill all the column along cos θ the series as 4, 3, 2, 1 and 0
3. 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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