Trigonometry is a discipline of mathematics that focuses on triangles, the relationship between side length and angles and their application using formulas, identities, and rules. Engineering, architecture, stringed musical instruments, and a variety of other scientific fields utilise trigonometric identities. 

It includes dealing with side ratios in triangles, which are utilised to calculate angle measurements. Sine and cosine are the most fundamental trigonometric functions. To use trigonometry formulas, it extends to many relationships between sine, cosine, tangent, secant, cosecant, and cotangent.

Only right-angled triangles must be considered when learning about trigonometric formulas. They can, however, be used on other triangles. The hypotenuse, the opposite side (perpendicular), and the adjacent side are the three sides of a right-angled triangle (base). The hypotenuse is the longest side of the triangle. 

The perpendicular is the side opposing the angle, and the adjacent side is the side on which both the hypotenuse and the opposite side rest.

To understand trigonometry basics, let us understand triangles and the relationship of their sides and angles.

AB = Perpendicular (opposite side)

BC = Base (adjacent side)

AC = Hypotenuse

Θ = theta

Basic trigonometric formulas:

In all trigonometric formulas, there are six fundamental trigonometric ratios. These ratios are also referred to as trigonometric functions, and they often employ all trigonometry formulas. Sine, cosine, secant, cosecant, tangent, and cotangent are the six essential trigonometric functions. 

The right-angled triangle is used to generate trigonometric functions and identities. We may use trigonometric formulas to get the sine, cosine, tangent, secant, cosecant, and cotangent values when we know the height and base side of the right triangle. 

Some relationships to keep in mind (reciprocal identities):

Sine = 1/Cosec                             Cos= 1/ Sec                                  Tan = 1/Cot

Cosec = 1/Sine                             Sec = 1/Cos                                  Cot = 1/ Tan

Sin θ Perpendicular                       AB

Hypotenuse                       AC

Cos θ Base                                     BC

Hypotenuse                       AC

Tan θ Perpendicular                   AB

Base                                  BC

Cot θ Base                                  BC

Perpendicular                AB                     

Sec θ Perpendicular                   AB

Base                                  BC

Cosec θ Hypotenuse                        AC

Perpendicular                     AB

Ratio tables in relation to angles in trigonometry

Trigonometry formulas and regularly used angles for solving trigonometric issues are shown in the table below. The trigonometric ratios table aids in determining the values of common trigonometric angles such as 0°, 30°, 45°, 60°, and 90°.

Angles (in degree) 30° 45° 60° 90° 180° 270° 360°
Angles (in radian) π/6 π/4 π/3 π/2 π 3π/2 2 π
Sin 0 ½ 1/√2 √3/2 1 0 -1 0
Cos 1 √3/2 1/√2 1/2 0 -1 0 1
Tan 0 1/√3 1 √3 0 0
Cot √3 1 1/√3 0 0
Cosec 2 √2 2/√3 1 -1
Sec 1 2/√3 √2 2 1 1

Triangles, light, sound, and waves all need the use of trigonometric functions. The following table shows the values of certain trigonometric functions in various domains and ranges:

Trigonometric Functions Domain Range
Sin a R -1 ≤ sin a ≤ 1
Cos a R -1 ≤ cos a ≤ 1
Tan a R – {(2n + 1)π/2, n I} R
Cosec a R – {nπ, n I} R – {a: -1 < a < 1}
Sec a R – {(2n + 1)π/2, n I} R – {a: -1 < a < 1}
Cot a R – {nπ, n I} R

Now that we have understood the angle relations, let us learn periodic formulas

These identities are useful to shift angles by π/2, π, 2π etc.

  1. sin(π/2–X) = cos X                                       cos(π/2–X) = sin X
  2. sin(π/2+X) = cosX                                        cos(π/2+X) = –sin X
  3. tan(π/2+X) = X                                             cot(π/2+X) = –tanX
  4. tan(π/2-X)=cot X                                           cot(π/2– X) =   tanX
  5. sin(π–X) = sinX                                              cos(π– X)  = –cosX
  6. sin(π+X)=sinX                                                cos(π+ X)  = –cosX
  7. tan(π+X)=tanX                                               cot(π+ X)   =  tanX
  8. tan(π–X)=–tanX                                             cot(π–X) =  –cotX
  9. sin(3π/2–X)= –cosX                                      cos(3π/2–X) = –sinX
  10. sin(3π/2+X)= –cosX                                      cos(3π/2+X) =  sinX
  11. tan(3π/2+X)= X                                             cot(3π/2+X) = –tanX
  12. tan(3π/2–X)= cot X                                       cot(3π/2–X) = tanX
  13. sin(2π–X) = sinX                                            cos(2π–X) =  cosX
  14. sin(2π+X) = sinX                                            cos(2π+X) = –cosX
  15. tan(2π+X) = tanX                                            cot(2π+X) = cot⁡X
  16. tan(2π–X) = –tanX                                           cot(2π–X)= –cotX

Co function identity for trigonometric functions

  1. cos(90°– a ) = sin a 
  2. tan(90°– a) = cot a 
  3. cot(90°– a) = tan a
  4. sec(90°– a)=  cosec a
  5. cosec(90°– a)= sec a

Sum and difference of identities for trigonometric functions

  1. sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
  2. cos(a+b) = cos(a)cos(b) – sin(a)sin(b)
  3. tan(a+b) = (tana+tanb)/(1–tana.tanb)
  4. sin(a–b) = sin(a)cos(b)–cos(a)sin(b)
  5. cos(a–b) =cos(a)cos(b)+sin(a)sin(b)
  6. tan(a–b) = (a–tanb)/(1+tana.tanb)

Half angle identity for trigonometric functions

  1. sin a/2 =   ± √ (1-cos a)/2
  2. cos a/2 = ± √ (1+cos a)/2
  3. tan a/2 = √ (1-cos a)/(1+cos a)

Double Angle Identities for trigonometric functions

  1. sin(2a) = 2sin(a).cos(a)
  2. cos(2a) = cos2a – sin2a = 1- 2sin2a = 2cos2a – 1
  3. tan(2a) = [2 tan(a)]/[1- tan2a]

Triple Angle Identities for trigonometric functions

  1. sin 3a =   3 sin a – 4 sin3 a
  2. cos 3a = 4 cos3a – 3 cos a
  3. tan 3a = [3 tan a – tan3 a]/ [1 – 3 tan2 a]

Square Law Formulas for trigonometric functions:

  1. sin 2a + cos 2a = 1
  2. tan 2a = 1 + sec 2a
  3. cot 2a = 1 + cosec 2a

One can solve any right triangle with the knowledge that the two sharp angles are complementary, i.e. they sum up to 90°:

We can find the third side and both acute angles if you know two of the three sides.

We can find the other acute angle and the other two sides if you know one acute angle and one of the three sides.

The sign of trigonometric functions is significant in their formulations because it varies when the quadrant changes. The sign is determined by the quadrant in which the angle is located.

Every trigonometric ratio in Q1 is positive. (Angles ranging from 0 to 90 degrees).

In Q2, all sin and cosec trigonometric ratios are positive. (Angles ranging from 90° to 180°)

In Q3, all cos and sec trigonometric ratios are positive. (Angles ranging from 180° to 270°).

In Q4, all tan and cot trigonometric ratios are positive. (Angles ranging from 270° to 360°)

The values of trigonometric functions change when the angles change, but they remain the same for 90° and 270°, and 180° and 360°, as we add or subtract from 90° and 270°, respectively.

Sec (90° + a ) = Cos a

Cot (90° – a ) = Cos a

Tan (90° + a ) = – Cot  a

Tan (90° – a ) = Cot  a

Sec (90° + a ) = Cosec a

Sec (90° + a ) = Cosec a

Sin (270° – a ) = – Cos a

Sin (270° – a ) = – Cos a

The following domain and range of inverse trigonometric identities can help you recall some crucial inverse trigonometry formulas:

Trigonometric Functions Domain Range
Sin-1 a −1,1 −π/2,π/2
Cos-1 a −1,1 0,π
Tan-1 a R −π/2,π/2
Cot-1 a R 0,π
Sec-1 a R – (-1,1) 0,π – π/2
Cosec-1 a R – (-1,1) −π/2, π/2 – 0

The above table of domain and range of trigonometric identities demonstrates that for a

 -1,1, the Sin-1 x has an infinite number of solutions, but only one value lies in the intervals -π/2,π/2, which is referred to as the principal value.