Section 7-3
1
Math 150 Lecture Notes
Double-Angle, Half-Angle, and Product-Sum Formulas
Double-Angle Formulas Half-Angle Formulas
sin 2x = 2 sin x cos x
2
cos1
2
sin xx −
±=
cos 2x = cos
2
x – sin
2
x
2
cos1
2
cos xx +
±=
= 1 – 2 sin
2
x
= 2 cos
2
x – 1
tan 2x =
x
x
2
tan
1
tan2
−
x
x
x
xx
cos
1
sin
sin
cos1
2
tan +
=
−
=
Determination of + or – sign depends on
quadrant of
2
x
Lowering Powers
2
2cos1
sin
2
x
x
−
=
2
2cos1
cos
2
x
x
+
=
x
x
x
2
cos
1
2cos1
tan
2
+
−
=
Product-to-Sum Formulas
sin u cos v = ½ [sin(u + v) + sin(u – v)]
cos u sin v = ½ [sin(u + v) – sin(u – v)]
cos u cos v = ½ [cos(u + v) + cos(u – v)]
sin u sin v = ½ [cos(u – v) – cos(u + v)]
Sum-to-Product Formulas
sin x + sin y = 2 sin
2
yx
+
cos
2
yx
−
sin x - sin y = 2 cos
2
yx
+
sin
2
yx
−
cos x + cos y = 2 cos
2
yx
+
cos
2
yx
−
cos x - cos y = -2 sin
2
yx
+
sin
2
yx
−
