Exam 3 Formulas: Sum and Difference: tan( + ) = tan

Exam 3 Formulas:
Sum and Difference:
tan 𝑥+tan 𝑦
tan(𝑥 + 𝑦) = 1−tan 𝑥 tan 𝑦
Double Angle:
2 tan 𝑥
tan 2𝑥 =
1−tan2 𝑥
Power Reducing:
1−cos 2𝑥
tan2 𝑥 =
1+cos 2𝑥
Half Angle:
𝑥
2
1+cos 𝑥
2
𝑥
1−cos 𝑥
cos ( ) = ±√
tan 𝑥−tan 𝑦
tan(𝑥 − 𝑦) = 1+tan 𝑥 tan 𝑦
𝑥
2
1−cos 𝑥
2
sin ( ) = ±√
sin 𝑥
tan (2) = ±√1+cos 𝑥 = 1+cos 𝑥 =
1−cos 𝑥
sin 𝑥
Product to Sum:
1
sin 𝑥 cos 𝑦 = 2 [sin(𝑥 + 𝑦) + sin(𝑥 − 𝑦)]
1
sin 𝑥 sin 𝑦 = [cos(𝑥 − 𝑦) − cos(𝑥 + 𝑦)]
2
Sum to Product:
𝑥+𝑦
𝑥−𝑦
sin 𝑥 + sin 𝑦 = 2 sin ( 2 ) cos ( 2 )
𝑥+𝑦
𝑥−𝑦
) cos ( )
2
2
cos 𝑥 + cos 𝑦 = 2 cos (
Law of Sines
sin 𝐴
sin 𝐵
sin 𝐶
=
=
𝑎
𝑏
𝑐
Law of Cosines
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
Polar Information
𝑥 = 𝑟 cos 𝜃
𝑦 = 𝑟 sin 𝜃
Complex Numbers in Polar Form
𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃)
𝑎 = 𝑟 cos 𝜃
DeMoivre’s Theorem
𝑧 𝑛 = 𝑟 𝑛 [cos(𝑛𝜃) + 𝑖 sin(𝑛𝜃)]
1
cos 𝑥 sin 𝑦 = 2 [sin(𝑥 + 𝑦) − sin(𝑥 − 𝑦)]
1
2
cos 𝑥 cos 𝑦 = [cos(𝑥 + 𝑦) + cos(𝑥 − 𝑦)]
𝑥+𝑦
𝑥−𝑦
) sin ( 2 )
2
𝑥+𝑦
𝑥−𝑦
−2 sin ( ) sin ( )
2
2
sin 𝑥 − sin 𝑦 = 2 cos (
cos 𝑥 − cos 𝑦 =
𝑟2 = 𝑥2 + 𝑦2
𝑏 = 𝑟 sin 𝜃
tan 𝜃 =
𝑟 2 = 𝑎2 + 𝑏 2
𝑦
𝑥
tan 𝜃 =
𝑏
𝑎