Tuesday, November 24, 2009

4.4 Compound Angle Formulas

Compound Angle Formulas

For Cos

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.

For Sin

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)

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