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Level 462

Trigonometric Identities II


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1
sin² α + cos² α
sec² α
1 + tan² α
csc² α
1 + cot² α
cos α
sin(π/2 - α)
sin α
cos(π/2 - α)
cot α
tan(π/2 - α)
tan α
cot(π/2 - α)
sec α
csc(π/2 - α)
csc α
sec(π/2 - α)
sin(-α)
-sin α
cos(-α)
cos α
tan(-α)
-tan α
cot(-α)
-cot α
csc(-α)
-csc α
sec(-α)
sec α
sin(α±β)
sin α cos β ± cos α sin β
cos(α±β)
cos α cos β ∓ sin α sin β
tan(α±β)
(tan α ± tan β)/(1 ∓ tan α tan β)
sin(2α)
2 sin α cos α
cos(2α)
cos² α - sin² α
tan(2α)
(2 tan α)/(1 - tan² α)
sin² α
(1 - cos(2α))/2
cos² α
(1 + cos(2α))/2
tan² α
(1 - cos(2α))/(1 + cos(2α))
sin α + sin β
2 sin ((α + β)/2) cos ((α - β)/2)
sin α - sin β
2 cos ((α + β)/2) sin ((α - β)/2)
cos α + cos β
2 cos ((α + β)/2) cos ((α - β)/2)
cos α - cos β
-2 sin ((α + β)/2) sin ((α - β)/2)
sin α sin β
1/2[cos(α - β) - cos(α + β)]
cos α cos β
1/2[cos(α - β) + cos(α + β)]
sin α cos β
1/2[sin(α + β) + sin(α - β)]
cos α sin β
1/2[sin(α + β) - sin(α - β)]
tan(x) =
sin(x) / cos(x) =
cot(x) =
cos(x) / sin(x) =
sec(x) =
1 / cos(x) =
csc(x) =
1 / sin(x) =
1 =
sin² (x) + cos² (x)
sin² (x) =
1 - cos² (x) =
cos² (x) =
1 - sin² (x) =
sec² (x) =
1 + tan² (x) =
csc² (x) =
1 + cot² (x) =
tan x (basic)
(sin x)/(cos x) = 1/cot x
1/csc x
sin x (basic)
1/sec x
cos x (basic)
cot x (basic)
(cos x)/(sin x) = 1/tan x
1/sin x
csc x (basic)
1/cos x
sec x (basic)
1
sin² x + cos² x (pythagorean)
sec² x
tan² x + 1 (pythagorean)
csc² x
1 + cot² x (pythagorean)
sin(-x) (odd)
-sin x
tan(-x) (odd)
-tan x
csc(-x) (odd)
-csc x
cot(-x) (odd)
-cot x
cos(-x) (even)
cos x
sec(-x) (even)
sec x
cos u
sin((π/2) − u) (cofunction)
cot u
tan((π/2) − u) (cofunction)
csc u
sec((π/2) − u) (cofunction)
sin u
cos((π/2) − u) (cofunction)
tan u
cot((π/2) − u) (cofunction)
sec u
csc((π/2) − u) (cofunction)
Cosine of a Sum
cos(α + β) = cos α cos β − sin α sin β
Cosine of a Difference
cos(α − β) = cos α cos β + sin α sin β
Sine of a Sum
sin(α + β) = sin α cos β + cos α sin β
Sine of a Difference
sin(α − β) = sin α cos β − cos α sin β
Tangent of a Sum
tan(α + β) = (tan α + tan β)/(1 − tan α tan β)
Tangent of a Difference
tan(α − β) = (tan α − tan β)/(1 + tan α tan β)
sin 2x (double-angle)
2 sin x cos x
cos 2x (double-angle)
cos² x − sin² x = 2 cos² x − 1 = 1 − 2 sin² x
tan 2x (double-angle)
(2 tan x)/(1 − tan² x)
sin(x/2) (half-angle)
±√((1 − cos x)/2)
cos(x/2) (half-angle)
±√((1 + cos x)/2)
tan(x/2) (half-angle)
±√((1 − cos x)/(1 + cos x)) = sin x/(1 + cos x) = (1 − cos x)/sin x
sin A cos B (product-to-sum)
(1/2)(sin(A + B) + sin(A − B))
sin A sin B (product-to-sum)
(1/2)(cos(A − B) − cos(A + B))
cos A sin B (product-to-sum)
(1/2)(sin(A + B) − sin(A − B))
cos A cos B (product-to-sum)
(1/2)(cos(A − B) + cos(A + B))
sin x + sin y (sum-to-product)
2 sin((x + y)/2)cos((x − y)/2)
sin x − sin y (sum-to-product)
2 cos((x + y)/2)sin((x − y)/2)
cos x + cos y (sum-to-product)
2 cos((x + y)/2)cos((x − y)/2)
cos x − cos y (sum-to-product)
-2 sin((x + y)/2)sin((x − y)/2)
Reduction Formula
If α is an angle in standard position whose terminal side contains (a,b), then for any real number x:
Oblique Triangle
Side a is opposite angle α, side b is opposite angle β, and side c is opposite angle γ (generally, c is the longest side)
Law of Sines
In any triangle, (sin α)/a = (sin β)/b = (sin γ)/c
Law of Cosines
a² = b² + c² − 2bc cos α
2sinAcosB
sin(A+B) + sin(A-B)
2cosAsinB
sin(A+B) - sin(A-B)
2cosAcosB
cos(A+B) + cos(A-B)
-2sinAsinB
cos(A+B) - cos(A-B)
Rsin(θ+α)
a sinθ + bcosθ
Rcos(θ+α)
a cosθ - b sinθ
R
√(a² + b²)
α
tan⁻¹ (b/a)
1
sin²x + cos²x
sec²x
1 + tan²x
cosec²x
1 + cot²x
sin2x
2sinxcosx
cos2x
cos²x - sin²x
tan2x
2tanx / 1 - tan²x
sin²(½)x
½(1 - cosx)
cos²(½)x
½(1 + cosx)