5.5 Multiple-Angle Formulas
Double Angle Formulas
sin 2u = 2sin u cos u
cos 2u = cos²u - sin²u
= 2cos² u - 1
= 1 - 2sin² u
tan 2u = (2tan u)/(1 - tan² u)
sin 2u = 2sin u cos u
cos 2u = cos²u - sin²u
= 2cos² u - 1
= 1 - 2sin² u
tan 2u = (2tan u)/(1 - tan² u)
Solving a Multiple Angle Equation
Evaluating Functions Involving Double Angles
The double angle formulas are not restricted to the angles 2θ and θ. Other double angle combinations, such as 4θ and 2θ or 6θ and 3θ, are also valid.
sin 4θ = 2 sin 2θ cos 2θ and cos 6θ = cos² 3θ - sin² 3θ
By using double angle formulas together with the sum formulas other multiple angle formulas can be formed.
sin 4θ = 2 sin 2θ cos 2θ and cos 6θ = cos² 3θ - sin² 3θ
By using double angle formulas together with the sum formulas other multiple angle formulas can be formed.
Deriving a Triple angle Formula
Power Reducing Formulas
The double-angle formulas can be used to obtain power reducing formulas.
sin² u = (1 - cos 2u)/2
cos² u = (1 + cos 2u)/2
tan² u = (1 - cos 2u)/(1 + cos 2u)
The double-angle formulas can be used to obtain power reducing formulas.
sin² u = (1 - cos 2u)/2
cos² u = (1 + cos 2u)/2
tan² u = (1 - cos 2u)/(1 + cos 2u)
Reducing a Power
Half- Angle Formulas
Some useful alternative forms of the power reducing formulas can be derived by replacing u with u/2.
sin (u/2) = ±√((1 - cos u)/2)
cos (u/2) = ±√((1 + cos u)/2)
tan (u/2) = (1 - cos u)/sin u = sin u /(1 + cos u)
The sings of sin (u/2) and cos (u/2) depend on the quadrant in which u/2 lies.
Some useful alternative forms of the power reducing formulas can be derived by replacing u with u/2.
sin (u/2) = ±√((1 - cos u)/2)
cos (u/2) = ±√((1 + cos u)/2)
tan (u/2) = (1 - cos u)/sin u = sin u /(1 + cos u)
The sings of sin (u/2) and cos (u/2) depend on the quadrant in which u/2 lies.
Using a Half Angle Formula
Solving a Trigonometric Equation
Product to Sum Formulas
sin u sin v = 1/2 [cos (u - v) - cos (u+ v)]
cos u cos v = 1/2 [cos (u -v) + cos (u + v)]
sin u cos v = 1/2[sin (u + v) + sin (u - v)]
cos u sin v = 1/2[sin (u + v) - sin(u - v)]
sin u sin v = 1/2 [cos (u - v) - cos (u+ v)]
cos u cos v = 1/2 [cos (u -v) + cos (u + v)]
sin u cos v = 1/2[sin (u + v) + sin (u - v)]
cos u sin v = 1/2[sin (u + v) - sin(u - v)]
Writing Products as Sums
Sum to Product Formulas
sin u + sin v = 2 sin((u + v)/2) cos ((u - v)/2)
sin u - sin v = 2 cos ((u + v)/2) sin ((u - v)/2)
cos u + cos v = 2 cos ((u + v)/2) cos ((u - v)/2)
cos u - cos v = -2 sin ((u + v)/2) sin ((u - v)/2)
sin u + sin v = 2 sin((u + v)/2) cos ((u - v)/2)
sin u - sin v = 2 cos ((u + v)/2) sin ((u - v)/2)
cos u + cos v = 2 cos ((u + v)/2) cos ((u - v)/2)
cos u - cos v = -2 sin ((u + v)/2) sin ((u - v)/2)
Using a Sum to Product Formula
Solving a Trigonometric Equation