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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