Download Trigonometric Identities Sum and Difference Formulas sin (x + y

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
q
θ
sin 2θ = ± 1−cos
2
tan 2θ =
1−cos x
sin x
tan (x − y) =
Half-Angle Formulas
q
θ
cos 2θ = ± 1+cos
2
tan 2θ =
tan x−tan y
1+tan x tan y
q
θ
tan 2θ = ± 1−cos
1+cos θ
sin θ
1+cos θ
Double-Angle Formulas
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ
cos 2θ = 2 cos2 θ − 1
cos 2θ = 1 − 2 sin2 θ
tan 2θ =
2 tan θ
1−tan2 θ
Product-to-Sum Formulas
sin x sin y = 12 [cos (x − y) − cos (x + y)]
cos x cos y = 21 [cos (x − y) + cos (x + y)]
sin x cos y = 12 [sin (x + y) + sin (x − y)]
Sum-to-Product Formulas
sin x + sin y = 2 sin x+y
cos x−y
sin x − sin y = 2 sin x−y
cos x+y
2
2
2
2
x−y
x+y
x−y
cos x + cos y = 2 cos x+y
cos
cos
x
−
cos
y
=
−2
sin
sin
2
2
2
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