Trigonometry Formulas and Identities

 Trigonometry:

Trigonometry is an important branch of Mathematics. The word Trigonometry has been derived from three Greek words Trie(three) , goni(angles) and Metron(measurement). 

Literally it means Measurement of triangle.

Uses of trigonometry:

It is considerably used in business , engineering , surveying , navigation , astronomy , physical and social sciences.

Units of measures of Angles:

Two rays with a common starting point form an  Angle. One of the ray of the angle is called initial side and other is called terminal side. An angle is said to be positive/negative if rotation is anticlockwise/clockwise.

 Angles are usually denoted by Greek letters such as Alpha(∝), beta(β), gamma(ɤ),  theta(θ) etc. 

There are two commonly used measurements for angle : 

Degree and Radians which are:

Sexagesimal system and circular system

       Sexagesimal system (Degree, minutes, seconds)

if the initial ray OA rotates in anticlockwise direction in such a way that it coincides with itself, the angle then formed is said to be 360 degree.

• One rotation (anticlockwise) = *360 degree
• 1/2 rotation (anticlockwise) = 180 degree is called a straight angle
• 1/4 rotation (anticlockwise) = 90 degree is called a right angle. 

1 degree is divided into 60 minutes (60') and 1 minute is divided into 60 seconds (60"). As this system of Measurement of the Angle owns its origin to the English and because 90 , 60 are multiple of 6 and 10 , so it is also known as English sexagesimal system.

Thus,

• 1 rotation = 360° (360 degree)
• One degree = 60 minutes
• One minute = 60 seconds

Circular system

There is another angular measurement , called the Circular system. It is most useful for the study of higher Mathematics. Specially calculus.

According to a proper definition:

Radian is the measure of the Angle subtended at the center of the circle by an arc , whose length is equal to its radius (radius of the circle).

Relation between the length of an arc of a circle and the circular measure of its central angle

if r is the radius , L is the length of the arc and theta is circular measure of the central angle.

By definition of Radian;

• ⇒An Angle of 1 radian subtended an arc AB on the circle of the length= 1.r
• ⇒An Angle of 1/2 radian subtended an arc AB on the circle of the length= 1/2.r
• ⇒An Angle of 2 radian subtended an arc AB on the circle of the length= 2.r
• ⇒An angle of theta Radian subtended an arc AB on the circle of the length= theta.r

• ⇒ AB = theta.r
• ⇒ L = theta.r
• theta = L/r
• so it is proved that theta = L/r

Trigonometric functions:

There are six Trigonometric functions:

1. Sineθ = a/c
2. Cosineθ = b/c
3. Tangentθ = a/b
4. cosecantθ = c/a
5. Secentθ = c/b
6. Cotangentθ = b/a

Relation between Trigonometric functions:

• Cscθ = 1/Sinθ 
• secθ = 1/cosθ
• tanθ = 1/tanθ
• tanθ = sin/cosθ
• cotθ = 1/tanθ

Fundamental identities:

fundamental identities are:

1. sinθ² + cosθ² = 1
2. 1 + tanθ² = sec²
3. 1+ cotθ² = csc²

Note that:

 (sinθ)² = (sinθ²)

(cosθ)² = (cosθ²)

(tanθ)² = (tanθ²)