Compound Angle Formulas
Cos(a-b) = Cos a Cos b + Sin a Sin b
Cos(a+b) = Cos a Cos b - Sin a Sin b
Notices that when the left half of the equation is a minus, the right half of the equation is plus. And vice versa.
Sin(a-b) = Sin a Cos b - Cos a Sin b
Sin(a+b) = Sin a Cos b + Cos a Sin b
And as for Sin the plus and minus sign stayed the same on both side. The difference with these two formulas is that here is Sin x Cos and Cos x Sin a mixture of sin and cos where as for Cos is only Cos x Cos and Sin x Sin.
We use these four formulas to determine the exact value fo an angle.
For example: cos( π/3 - π/4)
First we write out the whole equation:
Cos(π/3 - π/4) = Cos(π/3)Cos(π/4) + Sin(π/3)Sin(π/4)
Next we look at the exact value for Cos(π/3),Cos(π/4),Sin(π/3),Sin(π/4)
If we look at our special triangles we know that: Cos(π/3)is 1/2, Cos(π/4) is 1/√ 2, Sin(π/3) is √3/2, and Sin(π/4) is 1/√ 2, so now our new equation will be:
(1/2)(1/√ 2) + (√3/2)(1/√ 2)
Now we simplify it:
(1/2√ 2) + (√ 3/2√ 2)
which equals to:
1+√ 3/ 2√ 2
Another example to use these equations will be express the formulas as a single trigonometric function.
example: Express as a single trigonometric function Sinπ Cosπ/4 + Cosπ Sin π/4
Equals: Sin (π+ π/4)
= Sin (5π/4)