# Trigonometric ratios of some Specific Angles

Last Updated : 27 Mar, 2024

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,
Â Â

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).

[Tex]sin\, C = \frac{BA}{AC}Â Â Â Â [/Tex]Â Â

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

[Tex]cos\, C = \frac{BC}{AC}Â Â Â Â [/Tex]Â Â

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

[Tex]tan\, C = \frac{BA}{BC}Â Â Â Â [/Tex]Â Â

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).Â

[Tex]cosec\, C = \frac{AC}{BA}Â Â Â Â [/Tex]Â Â

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).Â

[Tex]sec\, C = \frac{AC}{BC}Â Â Â Â [/Tex]Â Â

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).Â

[Tex]cot\, C = \frac{BC}{BA}[/Tex]

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

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 –Â

if A = 0Â° Â Â [Tex]\\ sin A = \frac{BC}{AC} = \frac{b}{a} = 0 \\\quad\\ cos A = \frac{AB}{AC} = \frac{c}{a} =1 \\\quad\\ tan A = \frac{BC}{AB} = \frac{b}{a} = 0 \\\quad\\ cosec A = \frac{AC}{BC} = \frac{a}{b} = not\, defined \\\quad\\ sec A = \frac{AC}{AB} = \frac{a}{c}= 1 \\\quad\\ cot A = \frac{AB}{BC} = \frac{a}{b}= not\, defined \\\quad\\[/Tex]

if A = 90Â° Â Â [Tex]\\ sin A = \frac{BC}{AC} = \frac{b}{a} = 1 \\\quad\\ cos A = \frac{BA}{AC} = \frac{c}{a} = 0 \\\quad\\ tan A = \frac{BC}{BA} = \frac{b}{c} = not\, defined \\\quad\\ cosec A = \frac{AC}{BC} = \frac{b}{a}= 1 \\\quad\\ sec A = \frac{AC}{BA} = \frac{a}{c}= not\, defined \\\quad\\ cot A = \frac{BA}{BC} = 0[/Tex]

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°,Â

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

[Tex]sin\ A = \frac{BC}{AC} = \frac{a}{a\sqrt2} = \frac{1}{\sqrt2}\\ \quad\\ cos\ A = \frac{AB}{AC} = \frac{a}{a\sqrt2} = \frac{1}{\sqrt2}\\ \quad\\ tan\ A = \frac{BC}{AB} = \frac{a}{a} = 1\\ \quad\\ cosec\ A = \frac{1}{sin\ A}= \sqrt2\\ \quad\\ sec\ A = \frac{1}{cos\ A} = \sqrt2\\ \quad\\ cot\ A = \frac{1}{tan\ A} = 1\\[/Tex]

Previous
Next