Trigonometry is all about triangles or to more precise about the relation between the angles and sides of a right-angled triangle. In this article we will be discussing about the ratio of sides of a right-angled triangle respect to its acute angle called trigonometric ratios of the angle and find the trigonometric ratios of specific angles: 0°, 30°, 45°, 60°, and 90°.

Consider the following triangle:

The side BA is opposite to angle ∠BCA so we call BA the opposite side to ∠C and AC is the hypotenuse, the other side BC is the adjacent side to ∠C.

## Trigonometric Ratios of angle C

Sine:Sine of ∠C is the ratio between BA and AC that is the ratio between the side opposite to C and the hypotenuse.

Cosine:Cosine of ∠C is the ratio between BC and AC that is the ratio between the side adjacent to C and the hypotenuse.

Tangent:Tangent of ∠C is the ratio between BA and BC that is the ratio between the side opposite and adjacent to C

Cosecant:Cosecant of ∠C is the reciprocal of sin C that is the ratio between the hypotenuse and the side opposite to C.

Secant:Secant of ∠C is the reciprocal of cos C that is the ratio between the hypotenuse and the side adjacent to C.

Cotangent:Cotangent of ∠C is the reciprocal of tan C that is the ratio between the side adjacent to C and side opposite to C.

## Finding trigonometric ratios for angle 0°, 30°, 45°, 60°, 90°

### A. For angles **0° and 90**°

If an angle A = 0° then the length of the opposite side would be zero and hypotenuse = adjacent side and if A = 90° then the hypotenuse = opposite side. So by using the above formulas for the trigonometric ratios and if the length of the hypotenuse is **a**.

if A = 0°

if A = 90°

Here csc 0, cot 0, tan 90 and sec 90 are not defined as at the particular angle it is divided by 0 which is undefined.

### B. For angles **30° and 60**°

Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore,

∠A = ∠B = ∠C = 60°.

∆ABD is a right triangle, right-angled at D with ∠BAD = 30° and ∠ABD = 60°,

Here ∆ADB and ∆ADC are similar as they are Corresponding parts of Congruent triangles(CPCT).

Now we know the values of AB, BD, and AD, So the trigonometric ratios for angle 30 are

**For angle 60°**

**C. For angle 45°**

In a right-angled triangle if one angle is 45° then the other angle is also 45° thus making it an isosceles right angle triangle

If the length of side BC=a then length of AB=a and Length of AC(hypotenuse) is a√2, then

### Result:

∠A |
0° |
30° |
45° |
60° |
90° |

sin A |
0 | 1/2 | 1/√2 | √3/2 | 1 |

cos A |
1 | √3/2 | 1/√2 | 1/2 | 0 |

tan A |
0 | 1/√3 | 1 | √3 | Not defined |

sec A |
Not defined | 2 | √2 | 2/√3 | 1 |

csc A |
1 | 2/√3 | √2 | 2 | Not defined |

cot A |
Not defined | √3 | 1 | 1/√3 | 0 |