Trigonometric Identities - Miami
[Pages:1]Trigonometric Identities
Sum and Difference Formulas
sin (x + y) = sin x cos y + cos x sin y
sin (x - y) = sin x cos y - cos x sin y
cos (x + y) = cos x cos y - sin 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
tan (x - y)
=
tan x-tan y 1+tan x tan y
Half-Angle Formulas
sin
2
=
?
1-cos 2
tan
2
=
1-cos x sin x
cos
2
=
?
1+cos 2
tan
2
=
sin 1+cos
Double-Angle Formulas
tan
2
=
?
1-cos 1+cos
sin 2 = 2 sin cos cos 2 = 2 cos2 - 1
cos 2 = cos2 - sin2 cos 2 = 1 - 2 sin2
tan 2
=
2 tan 1-tan2
Product-to-Sum Formulas
sin x sin y
=
1 2
[cos (x - y)
-
cos (x
+
y)]
cos x cos y
=
1 2
[cos (x
-
y)
+
cos (x
+
y)]
sin x cos y
=
1 2
[sin (x +
y) +
sin (x
-
y)]
Sum-to-Product Formulas
sin x + sin y = 2 sin
x+y 2
cos
x-y 2
sin x - sin y = 2 sin
x-y 2
cos
x+y 2
cos x + cos y = 2 cos
x+y 2
cos
x-y 2
cos x - cos y = -2 sin
x+y 2
sin
x-y 2
The Law of Sines
sin A sin B sin C
=
=
a
b
c
Suppose you are given two sides, a, b and the angle A opposite the side A. The height of the triangle is h = b sin A. Then
1. If a < h, then a is too short to form a triangle, so there is no solution.
2. If a = h, then there is one triangle.
3. If a > h and a < b, then there are two distinct triangles.
4. If a b, then there is one triangle.
The Law of Cosines
a2 = b2 + c2 - 2bc cos A
b2 = a2 + c2 - 2ac cos B
c2 = a2 + b2 - 2ab cos C
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