TRIGONOMETRY LAWS AND IDENTITIES - California State University San Marcos
TRIGONOMETRY
QUOTIENT IDENTITIES
tan(x)
=
sin(x) cos(x)
cot(x)
=
cos(x) sin(x)
RECIPROCAL IDENTITIES
csc(x) = 1 sin(x)
sec(x) = 1 cos(x)
cot(x) = 1 tan(x)
sin(x) = 1 csc(x)
cos(x) = 1 sec(x)
tan(x) = 1 cot(x)
EVEN/ODD IDENTITIES
sin(-x) = - sin(x) cos(-x) = cos(x) tan(-x) = - tan(x)
csc(-x) = - csc(x) sec(-x) = sec(x) cot(-x) = - cot(x)
PYTHAGOREAN IDENTITIES
cos2(x) + sin2(x) = 1 tan2(x) + 1 = sec2(x) cot2(x) + 1 = csc2(x)
SUM IDENTITIES
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
cos(x + y) = cos(x) cos(y) - sin(x) sin(y)
tan(x
+
y)
=
tan(x) + tan(y) 1 - tan(x) tan(y)
DIFFERENCE IDENTITIES
sin(x - y) = sin(x) cos(y) - cos(x) sin(y)
cos(x - y) = cos(x) cos(y) + sin(x) sin(y)
tan(x
-
y)
=
tan(x) - tan(y) 1 + tan(x) tan(y)
LAW OF SINES
sin(A) a
=
sin(B) b
=
sin(C ) c
csusm.edu/stemsc
XXX
LAWS AND IDENTITIES
DOUBLE-ANGLE IDENTITIES
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2(x) - sin2(x)
= 2 cos2(x) - 1
= 1 - 2 sin2(x)
tan(2x)
=
1
2 tan(x) - tan2(x)
HALF-ANGLE IDENTITIES
sin
x
=
r
?
1
-
cos(x)
cos
2 x
=
r
?
1
+
2 cos(x)
2
s 2
x
1 - cos(x)
tan
= ?
2
1 + cos(x)
PRODUCT TO SUM IDENTITIES
sin(x) sin(y)
=
1 2
[cos(x
-
y) - cos(x
+
y)]
cos(x) cos(y) = 1 [cos(x + y) + cos(x - y)] 2
sin(x) cos(y)
=
1 2
[sin(x
+
y)
+
sin(x
-
y)]
cos(x) sin(y)
=
1 2
[sin(x
+
y)
-
sin(x
-
y)]
SUM TO PRODUCT IDENTITIES
sin(x) + sin(y) = 2 sin x + y cos x - y
sin(x) cos(x) cos(x)
+ -
sin(y) cos(y) cos(y)
= = =
2 cos
2 cos
-2 sin
x x
2 + y 2 + y 2 x +
2
y sciosnsinxxx--222-2yyy
LAW OF COSINES
a2 = b2 + c2 - 2bc cos(A) b2 = a2 + c2 - 2ac cos(B) c2 = a2 + b2 - 2ab cos(C)
@csusm_stemcenter
Tel: STEM SC (N): (760) 750-4101 STEM SC (S): (760) 750-7324
TRIGONOMETRY
DEFINITIONS
sin(x) = Opposite csc(x) = Hypotenuse
H ypotenuse
Opposite
cos(x) = Adjacent sec(x) = Hypotenuse
H ypotenuse
Adjacent
tan(x) = Opposite Adjacent
cot(x) = Adjacent Opposite
Opposite
Hypotenuse
x Adjacent
y
y = sin(x)
1
3
2
2
1
2 x
y
y = cos(x)
1
3
2
2
1
y
y = tan(x)
1
3
2
2
1
2
x
2
LAWS AND IDENTITIES
y
p
-
1 2
,
3 2
p p
-
2 2
,
2 2
p
-
3 2
,
1 2
2
3
3
4
120
5
6
150 135
(0, 1)
2
90
p
1 2
,
3 2
p p
2 2
,
2 2
3
60
4
6
45 30
p
3 2
,
1 2
(-1, 0)
180
(1, 0)
3600 2
x
210 225
315 330
7
11
6
6
p
-
3 2
,
-
1 2
5 4
240
270
300
7 4
4
5
p
3 2
,
-
1 2
p p
3
3
3
p p
-
2 2
,
-
2 2
2
2 2
,
-
2 2
p
-
1 2
,
-
3 2
(0, -1)
p
1 2
,
-
3 2
y
y = csc(x)
1
2
1
3
2
2
y
y = sec(x)
1
3
2
2
2
1
y
y = cot(x)
1
2
1
3
2
2
csusm.edu/stemsc
XXX
csusm_stemcenter
Tel: STEM SC (N): (760) 750-4101 STEM SC (S): (760) 750-7324
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