Trigonometry Formulas

All trigonometric formulas are divided into two major systems: The Trigonometric Identities and Trigonometric Ratios.

Trigonometric Identities are some formulas that involve the trigonometric functions. These trigonometry identities are true for all values of the variables.

Trigonometric Ratio is known for the relationship between measurement of the angles and the length of the side of the right triangle.

Trigonometry Functions:

In a right angled triangle, we have basically 3 sides namely – Hypotenuse, Opposite side and Adjacent side. The longest side is known as the hypotenuse, the side opposite to hypotenuse is opposite and the side where both hypotenuse and opposite rests is the adjacent side.

There are basically 6 Laws 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 the above right angled triangle as reference, the trigonometric functions or trigonometric identities are derived:

Sine=OppositeHypotenuse

Secant=HypotenuseAdjacent

Cosine=AdjacentHypotenuse

Tangent=OppositeAdjacent

Co−Secant=HypotenuseOpposite

Co−Tangent=AdjacentOpposite

The reciprocal identities are given as:

CosecΘ=1sinΘ

secΘ=1cosΘ

cotΘ=1tanΘ

sinΘ=1CosecΘ

cosΘ=1secΘ

tanΘ=1cotΘ

All these are taken from a right angled triangle. With the length 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.

Basic Trigonometry Formulas:

A.Trigonometry Formulas involving Periodicity Identities:

• sin(x+2π)=sinx
• cos(x+2π)=cosx
• tan(x+π)=tanx
• cot(x+π)=cotx

All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identity.

tan 45 = tan 225  but this is true for cos 45 and cos 225.

Refer to the above trigonometry table to verify the values.

B.Trigonometry Formulas involving Cofunction Identities – degree:

• sin(90∘−x)=cosx
• cos(90∘−x)=sinx
• tan(90∘−x)=cotx
• cot(90∘−x)=tanx

C.Trigonometry Formulas involving 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)=tanx+tany1−tanx⋅tany
• 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)=tanx−tany1+tanx⋅tany

D.Trigonometry Formulas involving Double Angle Identities:

• sin(2x) = 2 sin(x).cos(x)
• cos(2x)=cos2(x)–sin2(x),
• cos(2x)=2cos2(x)−1
• cos(2x)=1–2sin2(x)
• tan(2x)=[2tan(x)][1−tan2(x)]

E.Trigonometry Formulas involving 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)

F.Trigonometry Formulas involving Product identities:

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

G.Trigonometry Formulas involving Sum to Product Identities:

• sinx+siny=2sinx+y2cosx−y2
• sinx−siny=2cosx+y2sinx−y2
• cosx+cosy=2cosx+y2cosx−y2
• cosx−cosy=−2sinx+y2sinx−y2

Trigonometry Class 11 Formulas

sin(−θ)=−sinθ

cos(−θ)=cosθ

tan(−θ)=−tanθ

cosec(−θ)=−cosecθ

sec(−θ)=secθ

cot(−θ)=−cotθ

Product to Sum Formulas

sinx siny=12[cos(x–y)−cos(x+y)]

cosxcosy=12[cos(x–y)+cos(x+y)]

sinxcosy=12[sin(x+y)+sin(x−y)]

cosxsiny=12[sin(x+y)–sin(x−y)]

Sum to Product Formulas

sinx+siny=2sin(x+y2)cos(x−y2)

sinx−siny=2cos(x+y2)sin(x−y2)

cosx+cosy=2cos(x+y2)cos(x−y2)

cosx−cosy=–2sin(x+y2)sin(x−y2)

Basic Formulas