Double angle formulae

Double angle formulae explained, questions referenced from c3&c4 book

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  • Created on: 24-06-10 01:28
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W6.1 ­ Double Angle Formulae
In this session, we return to the compound angle formulae from last week.
We'll take the versions which are adding A and B
and we'll replace B with another A.
Let's do it first for sine:
sin(A + B) = sin A cos B + cos A sin B
If we put B = A, this gives:
sin(A + A) = sin A cos A + cos A sin A
or: sin 2A = 2 sin A cos A.
You need to set out to remember this formula
and the other two we shall derive soon.
The next easiest to do is for tangent.
We have: tan (A + B) =
Putting B = A gives us: tan (A + A) =
or: tan 2A =
The cosine double angle formula gives us a bit more fun.
We start with: cos(A + B) = cos A cos B sin A sin B
and put B = A, giving:
cos(A + A) = cos A cos A sin A sin A
or: cos 2A = .........................................(i)
That's the basic cosine double angle formula.
Last year, we learnt the relationship: + = 1 ..............(ii)
which gives us: =1
Use this in (i) to substitute for and you get:
cos 2A = 1 2
If we go back to (ii) and change its subject to ,
substituting that in (i) gives:
cos 2A = 2 1
So there are three versions of the cosine double angle formula
and you need to know all three of them.
NB
The first version of cos2A is in the formula book,
but you have to remember the other two.
[You should, of course, remember all of them!]
Ex
Use sin2A to find sin120°.
If we put A = 60° we have:
CT NHC 22/02/2010

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°) = 2 sin 60° cos 60°
The magic triangle gives us sin 60° = and cos 60° =
so sin 120° = 2 =
We could, of course, get the same result from All, Sin, Tan, Cos
which tells us to look for a positive sine in the second quadrant.
We can also make use of the double angle formulae in proving trig identities.…read more

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Q.E.D.
The other thing you can be asked to do is to solve a trig equation which needs
you to use one of the double angle formulae.
Ex
Solve sin 2x = cos x, 0 x 360°
First of all, use the sine double angle formula on the LHS:
2 sin x cos x = cos x
We can now divide both sides by cos x,
unless, of course, cos x = 0.…read more

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The integrand appears in the RHS of one form of the expansion of cos 6x:
cos 6x = 1 2
So = (1 ­ cos 6x)
That again is something we learnt to integrate in Core 3.
Notice that, if the integrand was , you would need to use another
form of the cosine double angle formula:
cos 2x = 2 1
Drill
Ex 6c
Review Test
CD8 Parametric differentiation
W6.3 ­ Adding two sine waves
We'll start with an example.…read more

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Equating coefficients of sin : 3 = R cos ..................(ii)
Equating coefficients of cos : 4 = R sin ...................(iii)
(ii) and (iii) are a pair of simultaneous equations in R and ,
but you probably haven't come across a pair quite like them before.
We need a specialised method of solving them.
First, square and add the two equations:
(ii)2 + (iii)2 9 + 16 = +
i.e.…read more

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