FlashMath | trigonometry

fundamental trigonometric identities

sin²θ + cos²θ =
1
1 + tan²θ =
sec²θ
1 + cot²θ =
csc²θ
sin(-θ) =
-sinθ
cos(-θ) =
cosθ
tan(-θ) =
-tanθ

reciprocal identities

sinθ =
1/cscθ
cosθ =
1/secθ
tanθ =
sinθ/cosθ
cotθ =
cosθ/sinθ
secθ =
1/cosθ
cscθ =
1/sinθ

angle sum and difference identities

sin(A ± B) =
sinAcosB ± cosAsinB
cos(A ± B) =
cosAcosB ∓ sinAsinB
tan(A ± B) =
(tanA ± tanB) / (1 ∓ tanAtanB)
sin(2θ) =
2sinθcosθ
cos(2θ) =
cos²θ - sin²θ
tan(2θ) =
2tanθ / (1 - tan²θ)

product-to-sum identities

sinAcosB =
1/2[sin(A + B) + sin(A - B)]
cosAcosB =
1/2[cos(A + B) + cos(A - B)]
sinAsinB =
1/2[cos(A - B) - cos(A + B)]

sum-to-product identities

sinA + sinB =
2sin[(A + B)/2]cos[(A - B)/2]
cosA + cosB =
2cos[(A + B)/2]cos[(A - B)/2]
sinA - sinB =
2cos[(A + B)/2]sin[(A - B)/2]

half-angle identities

sin²(θ/2) =
(1 - cosθ) / 2
cos²(θ/2) =
(1 + cosθ) / 2
tan²(θ/2) =
(1 - cosθ) / (1 + cosθ)
sin(θ/2) =
±√[(1 - cosθ) / 2]
cos(θ/2) =
±√[(1 + cosθ) / 2]
tan(θ/2) =
sinθ / (1 + cosθ)

triple angle identities

sin(3θ) =
3sinθ - 4sin³θ
cos(3θ) =
4cos³θ - 3cosθ
tan(3θ) =
(3tanθ - tan³θ) / (1 - 3tan²θ)
sin(3A) =
3sinA - 4sin³A
cos(3A) =
4cos³A - 3cosA
tan(3A) =
(3tanA - tan³A) / (1 - 3tan²A)

even-odd identities

sin(-θ) =
-sinθ
cos(-θ) =
cosθ
tan(-θ) =
-tanθ
cot(-θ) =
-cotθ
sec(-θ) =
secθ
csc(-θ) =
-cscθ

miscellaneous identities

sin(π/2 - θ) =
cosθ
cos(π/2 - θ) =
sinθ
tan(π/2 - θ) =
cotθ
cot(π/2 - θ) =
tanθ
sec(π/2 - θ) =
cscθ
csc(π/2 - θ) =
secθ


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