Trigonometric ratios are written as the ratios of two sides with respect to an acute angle of a right angled triangle. “T-ratios” is another acronym used for the word trigonometric ratios.
Let’s start learning about a right angled triangle and then move on to its trigonometric ratios.
What is a right angled triangle?
A right angled triangle is a triangle which has one angle of measure 90°, out of its three angles. The other two angles of the triangle are always acute angles.
The name of a right angled triangle is written by placing the vertex with 90° at the middle of the other two vertices.
In the above diagram, ΔABC is a right angled triangle with vertex B with an angle of 90°.
Here, ∠ABC or ∠B = 90°
Also, ∠A and ∠C are acute angles.
Sides of right angled triangle
The three sides of a right angled triangle are named as base, perpendicular and hypotenuse.
- Base: Base is the bottom side of the triangle.
- Perpendicular: Perpendicular is the side of the right angled triangle which lies at 90° to the base.
- Hypotenuse: The side of the right angled triangle which lies directly opposite to its angle of 90°.
Again, in the above diagram, the right angled triangle ΔABC has three sides AB, BC and AC.
The side AB is the base of the triangle, which lies at the bottom of ΔABC, which is called the base
of ΔABC.
The side BC is at 90° to the base AB, which is called the perpendicular of ΔABC.
The side AC is opposite to ∠B, which is called as hypotenuse of ΔABC.
The angle formed between base and the perpendicular is always 90°.
Hypotenuse is the longest side than base and perpendicular in a right angled triangle.
What are trigonometric ratios?
Trigonometric ratios are the ratios of two sides of a right angled triangle. These ratios are calculated with respect to an acute angle in the triangle. There are six types of trigonometric ratios which are listed in the following table by their full and short names.
| S.N. | Trigonometric ratio name | Abbreviation |
|---|---|---|
| 1 | Sine | Sin |
| 2 | Cosine | Cos |
| 3 | Tangent | Tan |
| 4 | Cosecant | Cosec |
| 5 | Secant | Sec |
| 6 | Cotangent | Cot |
In the right angled ΔABC in the above diagram, the trigonometric ratios w.r.t. acute angle ∠A are written as:
- Sine of ∠A as Sin A
- Cosine of ∠A as Cos A
- Tangent of ∠A as Tan A
- Cosecant of ∠A as Cosec A
- Secant of ∠A as Sec A
- Cotangent of ∠A as Cot A
Cosecant, secant and cotangent are also known as reciprocals of sine, cosine and tangent respectively.
How to calculate trigonometric ratios?
In the right angled ΔABC, we can calculate the trigonometric ratios w.r.t. acute
angle ∠A. To calculate them, first find out which are the opposite and adjacent sides to ∠A.
BC is opposite to ∠A
AB is adjacent to ∠A
AC is hypotenuse
Sine ratio
Sine of ∠A = Side opposite to ∠A Hypotenuse = BC AC
Here, BC is perpendicular and AC is hypotenuse. The ratio can also be written as:
Sin A =
Perpendicular
Hypotenuse
Cosine ratio
Cosine of ∠A = Side adjacent to ∠A Hypotenuse = AB AC
AB is base and AC is hypotenuse.
∴ Cos A =
Base
Hypotenuse
Tangent ratio
Tangent of ∠A = Side opposite to ∠A Side adjacent to ∠A = BC AB
Or ratio of tangent can be written as Tan A = Perpendicular Base
Cosecant ratio
Cosec A = 1 Sin ACosecant of ∠A = Hypotenuse Side opposite to ∠A = AC BC
Secant ratio
Secant of ∠A = Hypotenuse Side adjacent to ∠A = AC AB
Sec A = 1 Cos ACotangent ratio
Cot A = 1 Tan ACotangent of ∠A = Side adjacent to ∠A Side opposite to ∠A = AB BC
Cot A can also be written as: Cot A = Cos A Sin A
The relationship between Sin, Cos and Tan is Tan A = Sin A Cos A
In ΔABC, right angled at B, AB = 8cm, BC = 6cm. Find all six trigonometric ratios of angle
∠A.
In ΔABC,
BC = Perpendicular = 6cm
AB = Base = 8cm
AC = Hypotenuse = ?
Using Pythagoras theorem,
(AC)2 = (AB)2 + (BC)2
= (8)2 + (6)2
= 64 + 36
= 100
AC = 100cm or Hypotenuse = 100cm
Sin A = PH
= BCAC
= 610
Cos A = BH
= ABAC
= 810
Tan A =PB
= BCAB
= 68
Cosec A = HP
= ACBC
= 106
Sec A = HB
= ACAB
= 108
Cot A = BP
= ABBC
= 86
The values of trigonometric ratios of an angle do not vary with length of sides of a triangle, if the angle involved in the ratios remains same.
Multiplication of trigonometric ratios
When Sin, Cos and Tan are multiplied by Cosec, Sec and Cot respectively, the result obtained is always 1.
Sin multiplied by Cosec:
Sin A . Cosec A = 1
Cos multiplied by Sec:
Cos A . Sec A = 1
Tan multiplied by Cot:
Tan A . Cot A = 1
How to memorize trigonometric ratios?
Trigonometric ratios can be easily remembered using this visual.
To make it more easy to remember, this shorter form can also be used. It can be recalled as “SPH – CBH – TPB”.
