---

Trigonometric Identities and Formulas

---

Notes: sin²A = (sin A)²

Basic

tan-1 = arctan

sin-1 = arcsin

cos-1 = arccos

Reciprocal Identities

sin A = 1 csc A

csc A = 1 sin A

cos A = 1 sec A

sec A = 1 cos A

tan A = 1 cot A

cot A = 1 tan A

Quotient Identities

tan A = sin A cos A

cot A = cos A sin A

Pythagorean Identities

sin 2A + cos 2A = 1

1 + tan 2A = sec 2A

1 + cot 2A = csc 2A

Cofunction Identities

sin ( π 2 − A) = cos A	tan ( π 2 − A) = cot A	sec ( π 2 − A) = csc A
cos ( π 2 − A) = sin A	cot ( π 2 − A) = tan A	csc ( π 2 − A) = sec A

Negative Angle Identities (or Even and Odd Identities)
The cosine and secant functions are even.

cos(-A) = cos A , even function

sec(-A) = sec A , even function

The sine, cosecant, tangent, and cotangent functions are odd.

sin(-A) = -sin A , odd function

csc(-A) = -csc A , odd function

tan(-A) = -tan A , odd function

cot(-A) = -cot A , odd function

Addition Formulas (or Sum and Difference Formulas)

sin(A + B) = (sinA)(cosB) + (cosA)(sinB)	sin(A – B) = (sinA)(cosB) – (cosA)(sinB)
cos(A + B) = (cosA)(cosB) – (sinA)(sinB)	cos(A – B) = (cosA)(cosB) + (sinA)(sinB)
tan(A + B) = tanA + tanB 1 – (tanA)(tanB)	tan(A − B) = tanA − tanB 1 + (tanA)(tanB)
cot(A + B) = (cotA)(cotB) − 1 cotA + cotB	cot(A − B) = (cotA)(cotB) + 1 cotA − cotB

Double Angle Formulas

sin(2A) = (2)(sinA)(cosA)

cos(2A) = cos2 A – sin2 A = 2 cos2 A – 1 = 1 – 2 sin2 A

tan(2A) = 2 tanA 1 – tan2 A

Half Angle Formulas

sin A 2  =  ± √ 1 – cos A 2

cos A 2  =  ± √ 1 + cos A 2

tan A 2  =  ± √ 1 – cos A 1 + cos A   =   1 − cosA sin A   =   sinA 1 + cos A

Product Formulas
(2)(sinA)(cosB) = sin(A + B) + sin(A – B)
(2)(cosA)(cosB) = cos(A + B) + cos(A – B)
(2)(cosA)(sinB) = sin(A + B) – sin(A – B)
(2)(sinA)(sinB) = cos (A – B) – cos(A + B)

Factoring Formulas
sin A + sin B = 2 cos[(A – B)/2] sin[(A + B)/2]
sin A – sin B = 2 cos[(A + B)/2] sin[(A – B)/2]
cos A + cos B = 2 cos[(A + B)/2] cos[(A – B)/2]
cos A – cos B = -2 sin[(A + B)/2] sin[(A – B)/2]

Trigonometric Functions of Acute Angles
sin X = opp / hyp = a / c , csc X = hyp / opp = c / a
tan X = opp / adj = a / b , cot X = adj / opp = b / a
cos X = adj / hyp = b / c , sec X = hyp / adj = c / b ,

Trigonometric Functions of Arbitrary Angles
sin X = b / r , csc X = r / b
tan X = b / a , cot X = a / b
cos X = a / r , sec X = r / a

Sine and Cosine Laws in Triangles
In any triangle we have:
1 - The sine law
sin A / a = sin B / b = sin C / c
2 - The cosine laws
a² = b² + c² - 2 b c cos A
b² = a² + c² - 2 a c cos B
c² = a² + b² - 2 a b cos C

Sum to Product Formulas
cosX + cosY = 2cos[ (X + Y) / 2 ] cos[ (X – Y) / 2 ]
sinX + sinY = 2sin[ (X + Y) / 2 ] cos[ (X – Y) / 2 ]

Difference to Product Formulas
cosX – cosY = - 2sin[ (X + Y) / 2 ] sin[ (X – Y) / 2 ]
sinX – sinY = 2cos[ (X + Y) / 2 ] sin[ (X – Y) / 2 ]

Product to Sum/Difference Formulas
cosX cosY = (1/2) [ cos (X – Y) + cos (X + Y) ]
sinX cosY = (1/2) [ sin (X + Y) + sin (X – Y) ]
cosX sinY = (1/2) [ sin (X + Y) – sin[ (X – Y) ]
sinX sinY = (1/2) [ cos (X – Y) – cos (X + Y) ]

Difference of Squares Formulas
sin ²X – sin ²Y = sin(X + Y)sin(X - Y)
cos ²X – cos ²Y = - sin(X + Y)sin(X - Y)
cos ²X – sin ²Y = cos(X + Y)cos(X - Y)

Multiple Angle Formulas
sin(3X) = 3sinX - 4sin ³X
cos(3X) = 4cos ³X - 3cosX
sin(4X) = 4sinXcosX – 8sin ³XcosX
cos(4X) = 8cos ⁴X - 8cos ²X + 1

Power Reducing Formulas
sin ²A = 1/2 – (1/2)cos(2A)
cos ²A = 1/2 + (1/2)cos(2A)
sin ³A = (3/4)sinA – (1/4)sin(3A)
cos ³A = (3/4)cosA + (1/4)cos(3A)
sin ⁴A = (3/8) – (1/2)cos(2A) + (1/8)cos(4A)
cos ⁴A = (3/8) + (1/2)cos(2A) + (1/8)cos(4A)
sin ⁵A = (5/8)sinA – (5/16)sin(3A) + (1/16)sin(5A)
cos ⁵A = (5/8)cosA + (5/16)cos(3A) + (1/16)cos(5A)
sin ⁶A = 5/16 – (15/32)cos(2A) + (6/32)cos(4A) - (1/32)cos(6A)
cos ⁶A = 5/16 + (15/32)cos(2A) + (6/32)cos(4A) + (1/32)cos(6A)

Trigonometric Functions Periodicity
sin (X + 2π) = sin X , period 2π
cos (X + 2π) = cos X , period 2π
sec (X + 2π) = sec X , period 2π
csc (X + 2π) = csc X , period 2π
tan (X + π) = tan X , period π
cot (X + π) = cot X , period π