Trigonometric Identities - University of Wisconsin–La Crosse

Trigonometric Identities MTH 151

Reciprocal Identities

1 csc t =

sin t 1

sin t = csc t 1

sec t = cos t 1

cos t = sec t 1

cot t = tan t 1

tan t = cot t

Tangent and Cotangent

sin t tan t =

cos t cos t cot t = sin t Pythagorean Identities

sin2 t + cos2 t = 1

1 + cot2 t = csc2 t

tan2 t + 1 = sec2 t

Formulas for Negatives

(1)

sin(-t) = - sin t

(12)

(2)

cos(-t) = cos t

(13)

tan(-t) = - tan t

(14)

(3)

csc(-t) = - csc t

(15)

(4)

sec(-t) = sec t

(16)

cot(-t) = - cot t

(17)

(5)

(6)

Cofunction Formulas

sin - u = cos u (18)

2

(7)

cos - u = sin u (19)

2

(8)

sec - u = csc u (20)

2

csc - u = sec u (21)

(9)

2

(10)

tan - u = cot u (22) 2

(11)

cot - u = tan u (23)

2

Addition Formulas

sin(u + v) = sin u cos v + cos u sin v

(24)

cos(u + v) = cos u cos v - sin u sin v

(25)

tan u + tan v

tan(u + v) =

(26)

1 - tan u tan v

Subtraction Formulas

sin(u - v) = sin u cos v - cos u sin v

(27)

cos(u - v) = cos u cos v + sin u sin v

(28)

tan u - tan v

tan(u - v) =

(29)

1 + tan u tan v

Double-Angle Formulas

Half-Angle Formulas

sin 2u = 2 sin u cos u

(30)

cos 2u = cos2 u - sin2 u

(31)

cos 2u = 1 - 2 sin2 u

(32)

cos 2u = 2 cos2 u - 1

(33)

2 tan u

tan 2u = 1 - tan2 u

(34)

Half-Angle Identities

v

1 - cos v

sin = ?

(38)

2

2

v

1 + cos v

cos = ?

(39)

2

2

v

1 - cos v

tan = ?

(40)

2

1 + cos v

sin2 u = 1 - cos 2u

(35)

Alternate Half-Angle Formulas

2

cos2 u = 1 + cos 2u

(36)

2

tan2 u = 1 - cos 2u

(37)

1 + cos 2u

v 1 - cos v

tan =

(41)

2 sin v

v sin v

tan =

(42)

2 1 + cos v

Product-to-Sum Formulas

sin u cos v

=

1 2

[sin(u

+

v)

+

sin(u

-

v)]

(43)

cos u sin v

=

1 2

[sin(u

+

v)

-

sin(u

-

v)]

(44)

cos u cos v

=

1 2

[cos(u

+

v)

+

cos(u

-

v)]

(45)

sin u sin v

=

1 2

[cos(u

-

v)

-

cos(u

+

v)]

(46)

Sum-to-Product Formulas

a+b a-b

sin a + sin b = 2 sin

cos

(47)

2

2

a+b a-b

sin a - sin b = 2 cos

sin

(48)

2

2

a+b a-b

cos a + cos b = 2 cos

cos

(49)

2

2

a+b a-b

cos a - cos b = -2 sin

sin

(50)

2

2

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