Trigonometric Formulas

Posted by admin at October 18, 2020

Basic Formulas

There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, co-secant, tangent and co-tangent.

By using a right-angled triangle as a reference, the trigonometric functions or identities are derived:

  • sin θ = Opposite Side/Hypotenuse
  • cos θ = Adjacent Side/Hypotenuse
  • tan θ = Opposite Side/Adjacent Side
  • sec θ = Hypotenuse/Adjacent Side
  • cosec θ = Hypotenuse/Opposite Side
  • cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

The Reciprocal Identities are given as:

  • cosec θ = 1/sin θ
  • sec θ = 1/cos θ
  • cot θ = 1/tan θ
  • sin θ = 1/cosec θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ

All these are taken from a right angled triangle With the height and base side of the right triangle given, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.

Trigonometry Table

Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

Angles (In Degrees)30°45°60°90°180°270°360°
Angles (In Radians)π/6π/4π/3π/2π3π/2
sin01/21/√2√3/210-10
cos1√3/21/√21/20-101
tan01/√31√300
cot√311/√300
csc2√22/√31-1
sec12/√3√22-11

Periodicity Identities (in Radians)

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.

  • sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
  • sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
  • sin (3π/2 – A)  = – cos A & cos (3π/2 – A)  = – sin A
  • sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
  • sin (π – A) = sin A &  cos (π – A) = – cos A
  • sin (π + A) = – sin A & cos (π + A) = – cos A
  • sin (2π – A) = – sin A & cos (2π – A) = cos A
  • sin (2π + A) = sin A & cos (2π + A) = cos A

All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.

Co-function Identities (in Degrees)

The co-function or periodic identities can also be represented in degrees as:

  • sin(90°−x) = cos x
  • cos(90°−x) = sin x
  • tan(90°−x) = cot x
  • cot(90°−x) = tan x
  • sec(90°−x) = csc x
  • csc(90°−x) = sec x

Sum & Difference Identities

  • sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
  • cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
  • tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
  • sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
  • cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
  • tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

Double Angle Identities

  • sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
  • cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
  • cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
  • tan(2x) = [2tan(x)]/ [1−tan2(x)]
  • sec (2x) = secx/(2-sec2 x)
  • csc (2x) = (sec x. csc x)/2

Triple Angle Identities

  • Sin 3x = 3sin x – 4sin3x
  • Cos 3x = 4cos3x-3cos x
  • Tan 3x = [3tanx-tan3x]/[1-3tan2x]

Half Angle Identities

  • sinx2=±1−cosx2−−−−−−√
  • cosx2=±1+cosx2−−−−−−√
  • tan(x2)=1−cos(x)1+cos(x)−−−−−−√

Also, tan(x2)=1−cos(x)1+cos(x)−−−−−−√=(1−cos(x))(1−cos(x))(1+cos(x))(1−cos(x))−−−−−−−−−−−−−√=(1−cos(x))21−cos2(x)−−−−−−−−√=(1−cos(x))2sin2(x)−−−−−−−−√=1−cos(x)sin(x) So, tan(x2)=1−cos(x)sin(x)

Product identities

  • sinx⋅cosy=sin(x+y)+sin(xy)2
  • cosx⋅cosy=cos(x+y)+cos(xy)2
  • sinx⋅siny=cos(xy)−cos(x+y)2

Sum to Product Identities

  • sinx+siny=2sinx+y2cosxy2
  • sinx−siny=2cosx+y2sinxy2
  • cosx+cosy=2cosx+y2cosxy2
  • cosx−cosy=−2sinx+y2sinxy2

Inverse Trigonometry Formulas

  • sin-1 (–x) = – sin-1 x
  • cos-1 (–x) = π – sin-1 x
  • tan-1 (–x) = – tan-1 x
  • cosec-1 (–x) = – cosec-1 x
  • sec-1 (–x) = π – sec-1 x
  • cot-1 (–x) = π – cot-1 x


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