# Trigonometric ratios of some Specific Angles

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

Consider the following triangle,

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

### Trigonometric Ratios of angle C

Sine:Sine of âˆ C is the ratio of the side opposite to C (BA) to the hypotenuse (AC).

Cosine:Cosine of âˆ C is the ratio of the side adjacent to C (BC) and the hypotenuse (AC).

Tangent:The tangent of âˆ C is the ratio between the side opposite (BA) and adjacent to C (BC).

Cosecant:Cosecant of âˆ C is the reciprocal of sin C therefore it is the ratio of the hypotenuse (AC) to the side opposite to C (BA).

Secant:Secant of âˆ C is the reciprocal of cos C therefore it is the ratio of the hypotenuse (AC) to the side adjacent to C (BC).

Cotangent:Cotangent of âˆ C is the reciprocal of tan C that is the ratio of the side adjacent to C (BC) to the side opposite to C (BA).

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

Considering the length of the hypotenuse AC = a, BC = b and, BA = c.

**A. For angles 0Â° and 90Â°**

If angle A = 0Â°, the length of the opposite side would be zero and hypotenuse = adjacent side, and if A = 90Â°, the hypotenuse = opposite side. So, with the help of the above formulas for the trigonometric ratios we get –

if A = 0Â°

if A = 90Â°

Here some of the trigonometric ratios result as **not defined** as at the particular angle it is divided by 0 which is undefined.

**B. For angles 30Â° and 60Â°**

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

âˆ A = âˆ B = âˆ C = 60Â°.

âˆ†ABD is a right triangle, right-angled at D with âˆ BAD = 30Â° and âˆ ABD = 60Â°,

Here âˆ†ADB and âˆ†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 30Â°** are,

**For angle 60Â°**

**C. For angle 45Â°**

In a right-angled triangle if one angle is 45Â° then the other angle is also 45Â° 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 aâˆš2 using Pythagoras Theorem, then

### All values

âˆ A |
0Â° |
30Â° |
45Â° |
60Â° |
90Â° |

sin A |
0 | 1/2 | 1/âˆš2 | âˆš3/2 | 1 |

cos A |
1 | âˆš3/2 | 1/âˆš2 | 1/2 | 0 |

tan A |
0 | 1/âˆš3 | 1 | âˆš3 | Not defined |

cosec A |
Not defined | 2 | âˆš2 | 2/âˆš3 | 1 |

sec A |
1 | 2/âˆš3 | âˆš2 | 2 | Not defined |

cot A |
Not defined | âˆš3 | 1 | 1/âˆš3 | 0 |

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