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: 0°, 30°, 45°, 60°, and 90°.

Consider the following triangle,

The side BA is opposite to the 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 of the side opposite to C (BA) to the hypotenuse (AC).

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

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

Cosecant: Cosecant of ∠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 ∠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 ∠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 0°, 30°, 45°, 60°, 90° 

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

For angles 0° and 90°

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

if A = 0°

if A = 90°

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

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°

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 using Pythagoras Theorem, then

All Values of Trigonometric Ratios [Some Specific Angles]

Some of the common values of trigonometric ratios are listed in the following table:

∠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
cosec A	Not defined	2	√2	2/√3	1
sec A	1	2/√3	√2	2	Not defined
cot A	Not defined	√3	1	1/√3	0