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Trigonometry Functions Formulas

In a right-angled triangle, we have 3 sides namely – Hypotenuse, Opposite side (Perpendicular) and Adjacent side (Height). The longest side is known as the hypotenuse, the side opposite to the angle 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 a right angled triangle as reference, the trigonometric functions or trigonometric identities are derived:

sin θ = Opposite/Hypotenuse

cosec θ = Hypotenuse/Opposite

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 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.

Trigonometry Formulas List

A.Trigonometry Formulas involving Periodicity Identities:

sin(x+2π) = sin x

cos(x+2π) = cos x

tan(x+π) = tan x

cot(x+π) = cot x

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) = cos x

cos(90°−x) = sin x

tan(90°−x) = cot x

cot(90°−x) = tan x

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) = (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)

D.Trigonometry Formulas involving Double Angle Identities:

sin(2x) = 2sin(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)] = (sin 2x)/ (1–2sin2x)

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

And Many More….

Number of Pages:-21

General Science Notes