Trigonometry, as its name implies, is the study of triangles. It is an important branch of mathematics that studies the relationship between side lengths and angles of the right triangle and also aids in determining the missing side lengths or angles of a triangle. There are six trigonometric ratios or functions: sine, cosine, tangent, cosecant, secant, and cotangent, where cosecant, secant, and cotangent are the reciprocal functions of the other three functions, i.e., sine, cosine, and tangent, respectively. A trigonometric ratio is defined as the ratio of the side lengths of a right triangle. Trigonometry is employed in various fields in our daily life. It helps to determine the heights of hills or buildings. It is also used in fields like criminology, construction, physics, archaeology, marine engine engineering, etc.

Formulae of six trigonometric ratios/functions

Let us consider a right-angled triangle XYZ, where ∠Y = 90°. Let the angle at vertex Z be θ. The side adjacent to “θ” is called the adjacent side, and the side opposite to “θ” is called the opposite side. A hypotenuse is a side opposite to the right angle or the longest side of a right angle.

sin θ = Opposite side/Hypotenuse

cos θ = Adjacent side/Hypotenuse

tan θ = Opposite side/Adjacent side

cosec θ = 1/sin θ = Hypotenuse/Opposite side

sec θ = 1/ cos θ = Hypotenuse/Adjacent side

cot θ = 1/ tan θ = Adjacent side/Opposite side

Sine Formula

The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse to the given angle. A sine function is represented as “sin”.

sin θ = Opposite side/Hypotenuse

Cosine Formula

The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the hypotenuse to the given angle. A cosine function is represented as “cos”.

cos θ = Adjacent side/Hypotenuse

Some Basic Sine and Cosine Formulae

Sine and Cosine Functions in Quadrants

The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants.

The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants.

Degrees

Quadrant

Sign of Sine function

Sign of Cosine function

0° to 90°

1st quadrant

+ (positive)

+ (positive)

90° to 180°

2nd quadrant

+ (positive)

– (negative)

180° to 270°

3rd quadrant

– (negative)

– (negative)

270° to 360°

4th quadrant

– (negative)

+ (positive)

The negative angle identity of the sine and cosine functions

The sine of a negative angle is always equal to the negative sine of the angle.

sin (– θ) = – sin θ

The cosine of a negative angle is always equal to the cosine of the angle.

cos (– θ) = cos θ

Relation between sine and cosine function

sin θ = cos (90° – θ)

Reciprocal functions of the sine and cosine functions

A Cosecant function is the reciprocal function of the sine function.

cosec θ = 1/sin θ

A Secant function is the reciprocal function of the cosine function.

sec θ = 1/cos θ

Pythagorean identity

sin2θ + cos2θ = 1

Periodic identities of the sine and cosine functions

sin (θ + 2nπ) = sin θ

cos (θ + 2nπ) = cos θ

Double Angle formulae for the sine and cosine functions