Trigonometry Formulas

In Trigonometry Formulas, we will learn

• Basic Formulas

• sin, cos tan at 0, 30, 45, 60 degrees

• Pythagorean Identities

• Sign of sin, cos, tan in different quandrants

• Radians

• Negative angles (Even-Odd Identities)

• Value of sin, cos, tan repeats after 2π

• Shifting angle by π/2, π,  3π/2 (Co-Function Identities or Periodicity Identities)

• Angle sum and difference identities

• Double Angle Formulas

• Triple Angle Formulas

• Half Angle Identities (Power reducing formulas)

• Sum Identities (Sum to Product Identities)

• Product Identities (Product to Sum Identities)

• Law of sine

• Law of cosine

• What are Inverse Trigonometry Functions?

• Domain and range of Inverse Trigonometry Functions

• Inverse Trigonometric Formulas

• Inverse Trigonometry Substitutions

 

Basic Formulas

sin, cos tan at 0, 30, 45, 60 degrees

Pythagorean Identities

Signs of sin, cos, tan in different quadrants

To learn sign of sin, cos, tan in different quadrants,

we remember

A dd → S ugar → T o → C offee

 

 

Representing as a table

	Quadrant I	Quadrant II	Quadrant III	Quadrant IV
sin	+	+	–	–
cos	+	–	–	+
tan	+	–	+	–

 

Radians

Radian measure = π/180  ×  Degree measure

 

Also,

1 Degree = 60 minutes

i.e. 1° = 60’

 

1 Minute = 60 seconds

i.e. 1’ = 60’’

Negative angles (Even-Odd Identities)

sin (–x) = – sin x

cos (–x) = cos x

tan (–x) = – tan x

sec (–x) = sec x

cosec (–x) = – cosec x

cot (–x) = – cot x

 

Value of sin, cos, tan repeats after 2π

sin (2π + x) = sin x

cos (2π + x) = cos x

tan (2π + x) = tan x

Shifting angle by π/2, π,  3π/2 (Co-Function Identities or Periodicity Identities)

sin (π/2 – x) = cos x	cos (π/2 – x) = sin x
sin (π/2 + x) = cos x	cos (π/2 + x) = – sin x
sin (3π/2 – x)  = – cos x	cos (3π/2 – x)  = – sin x
sin (3π/2 + x) = – cos x	cos (3π/2 + x) = sin x
sin (π – x) = sin x	cos (π – x) = – cos x
sin (π + x) = – sin x	cos (π + x) = – cos x
sin (2π – x) = – sin x	cos (2π – x) = cos x
sin (2π + x) = sin x	cos (2π + x) = cos x

Angle sum and difference identities

Double Angle Formulas

Triple Angle Formulas

Half Angle Identities (Power reducing formulas)

Sum Identities (Sum to Product Identities)

Product Identities (Product to Sum Identities)

Product to sum identities are

  2 cos⁡x  cos⁡y = cos⁡ (x + y) + cos⁡(x - y)

  -2 sin⁡x  sin⁡y = cos⁡ (x + y) - cos⁡(x - y)

  2 sin⁡x  cos⁡y = sin⁡ (x + y) + sin⁡(x - y)

  2 cos⁡x  sin⁡y = sin⁡ (x + y) - sin⁡(x - y)

Law of sine

Here

• A, B, C are vertices of Δ ABC
• a is side opposite to A i.e. BC
• b is side opposite to B i.e. AC
• c is side opposite to C i.e. AB

Law of cosine

Just like Sine Law, we have cosine Law

What are Inverse Trigonometric Functions

If sin θ = x

Then putting sin on the right side

  θ = sin ^-1 x

  sin ^-1 x = θ

 

So, inverse of sin is an angle.

 

Similarly, inverse of all the trigonometry function is angle.

 

Note : Here angle is measured in radians, not degrees.

 

So, we have

  sin ^-1 x

  cos ^-1 x

  tan ^-1 x

  cosec ^-1 x

  sec ^-1 x

  tan ^-1 x

Domain and Range of Inverse Trigonometric Functions

	Domain	Range
sin -1	[–1, 1]	[-π/2,π/2]
cos -1	[–1, 1]	[0,π]
tan -1	R	(-π/2,π/2)
cosec -1	R – (–1, 1)	[π/2,π/2] - {0}
sec -1	R – (–1, 1)	[0,π]-{π/2}
cot -1	R	(0,π)

 

Inverse Trigonometry Formulas

Some formulae for Inverse Trigonometry are

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) = – sec ^-1 x

cot ^–1 (–x) = π – cot ^-1 x

 

Inverse Trigonometry Substitution