give me the formula of trigonometry
Answers
sin α, cos α
tan α = sin α, α ≠ π + πn, n є Zcos α2
cot α = cos α, α ≠ π + πn, n є Zsin α
tan α · cot α = 1
sec α = 1, α ≠ π + πn, n є Zcos α2
cosec α = 1, α ≠ π + πn, n є Zsin α
Pythagorean identity
sin2 α + cos2 α = 1
1 + tan2 α = 1cos2 α
1 + cot2 α = 1sin2 α
Sum-Difference Formulas
sin(α + β) = sin α · cos β + cos α · sin β
sin(α – β) = sin α · cos β – cos α · sin β
cos(α + β) = cos α · cos β – sin α · sin β
cos(α – β) = cos α · cos β + sin α · sin β
tan(α + β) = tan α + tan β1 – tanα · tan β
tan(α – β) = tan α – tan β1 + tanα · tan β
cot(α + β) = cotα · cot β - 1cot β + cot α
cot(α - β) = cotα · cot β + 1cot β - cot α
Double angle formulas
sin 2α = 2 sin α · cos α
cos 2α = cos2 α - sin2 α
tan 2α = 2 tan α1 - tan2 α
cot 2α = cot2 α - 12 cot α
Triple angle formulas
sin 3α = 3 sin α - 4 sin3 α
cos 3α = 4 cos3 α - 3 cos α
tan 3α = 3 tan α - tan3 α1 - 3 tan2 α
cot 3α = 3 cot α - cot3 α1 - 3 cot2 α
Power-reduction formula
sin2 α = 1 - cos 2α2
cos2 α = 1 + cos 2α2
sin3 α = 3 sin α - sin 3α4
cos3 α = 3 cos α + cos 3α4
Sum (difference) to product formulas
sin α + sin β = 2 sin α + β cos α - β22
sin α - sin β = 2 sin α - β cos α + β22
cos α + cos β = 2 cos α + β cos α - β22
cos α - cos β = -2 sin α + β sin α - β22
tan α + sin β = sin(α + β)cos α · cos β
tan α - sin β = sin(α - β)cos α · cos β
cot α + sin β = sin(α + β)sin α · sin β
cot α - sin β = sin(α - β)sin α · sin β
a sin α + b cos α = r sin (α + φ),
where r2 = a2 + b2, sin φ = b , tan φ = bra
Product to sum (difference) formulas
sin α · sin β = 1(cos(α - β) - cos(α + β))2
sin α · cos β = 1(sin(α + β) + sin(α - β))2
cos α · cos β = 1(cos(α + β) + cos(α - β))2
Tangent half-angle substitution
sin α = 2 tan (α/2)1 + tan2 (α/2)
cos α = 1 - tan2 (α/2)1 + tan2 (α/2)
tan α = 2 tan (α/2)1 - tan2 (α/2)
cot α = 1 - tan2 (α/2)2 tan (α/2)
By using a right-angled triangle as a reference, the trigonometric functions and 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 SideReciprocal 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
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 ASolved
Problems
Q.1:What is the value of (sin30° + cos30°) – (sin 60° + cos60°)?
Sol: Given,
(sin30° + cos30°) – (sin 60° + cos60°)
= ½ + √3/2 – √3/2 – ½
= 0
Q.2: If cos A = 4/5, then tan A = ?
Sol: Given,
Cos A = ⅘
As we know, from trigonometry identities,
1+tan2A = sec2A
sec2A – 1 = tan2A
(1/cos2A) -1 = tan2A
Putting the value of cos A = ⅘.
(5/4)2 – 1 = tan2 A
tan2A = 9/16
tan A = 3/4Inverse Trigonometry Formulas
sin-1 (–x) = – sin-1 x
cos-1 (–x) = π – cos-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