Up Learn – A Level maths (edexcel) – The Compound Angle Identities
Here’s a summary of everything you need to know about the compound angle identities – otherwise known as the compound angle formulae – for A Level.
More videos on The Compound Angle Identities:
Here’s a reminder of the key points you should know about the compound angle identities.
A compound angle is made up of multiple angles… either added together, or subtracted from each other.
And whenever we have a trig function with a compound angle for an input…
… It’s possible to rewrite that function, without the compound angle, using an identity.
[Stack the 3 identities vertically. Maybe show the LHSs on the first line of above and the RHSs on the 2nd..
tan(AB)=tan A tan B 1tan A tan B ]
The compound angle identity for sine is this… [emphasise sin(A±B)≡
The compound angle identity for cosine is this. [emphasise cos(AB)cos(A)cos(B)sin(A)sin(B)]
And the compound angle identity for tan, is this. [emphasise tan(AB)=tan A tan B 1tan A tan B ]
These identities are useful if we want to find new outputs from the trig functions, using outputs we already know.(wait for animation and pause for 2 sec) [Show this process using dynamic “drag and copy, then morph” approach with colour coding on the 30 and 45.sin(75)=sin 30+45 sin 30 cos 45 +cos 30 sin(45)=1222+3222=2+64]
They’re also useful for rewriting trig functions like this one, where the input is a compound angle and one part is unknown.(wait for animation and pause for 2 sec) [cos (x-45°)=cos(x)cos(45)+sin(x)sin(45)= 22cos (x) +22sin (x)]
For example, rewriting this function using a compound angle identity eventually allows us to solve the equation.. [Show sin (x+3) -cos (x) =0 from the start of the line and pulse box sin(x+pi/3) on “this”. Make sure the compound angle identity is also visible: sin(A±B)≡ABAB ]
We rewrite… [→ sin(x)cos(3) + cos(x)sin(3)-cos(x)=0 ]
… simplify… [12sin(x)+32cos (x)-cos(x)=0 ][12 sin(x)+3-22cos(x)=0]
And then use the tangent identity.(click once and wait for animation) [sin(x)cos(x)tan(x)] [Then show the steps to solving but can just bring them up one by one, no fancy animations needed: 12 sin(x)=-3-22cos(x) –>sin(x)=-(3-2)cos(x) –>sin(x)=(2-3)cos(x) –> sin(x)cos(x)=(2-3) –> tan(x)=(2-3) –> x=tan-1(2-3)=12 ]
And many equations can be solved this same way.
Next, there’s a whole set of other equations we can solve by using the compound angle identities in a different way.
In this case, this pesky term prevents us from dividing sin by cos and getting just one tan function like before. [Show trying to do this and show that we get 6tan(x) + 7 = 5/cos(x). Show that the 5/os(x) is a problem maybe with a wobbling ? over that term or a big cross or something]
Instead, we use a crafty trick: we use a compound angle identity in reverse.
[6sin x +7cos x =5, 0≤x≤180°] [Full working:
First, whenever we have some amount of sin(x) add some amount of cos(x), it’s possible to rewrite that as a single trig function with a compound input.
So we do that, putting an unknown here and here.
Second, we use this identity to rewrite this again.
Third, we equate the coefficients of sine x (wait 1 sec) and cos x.(wait 1 sec) [Step 3]
Fourth, we solve these simultaneous equations. [Step 4]
To find alpha, we divide the equations and then use the tangent identity. [tan(θ)=sin(θ)/cos(θ)]
And to find an exact value for R, we square these two values, add them and square root. [The 6 and the 7 in step 3]
Fifth, we put our values for R and alpha back into this equation. [Rsin(x+alpha)=5]
And now we’ve rewritten the left hand side in this form as planned. [Rsin(x+alpha)]
To finish, we solve as normal using the inverse trig function, making sure to find all the values in the range.(click once and pause for 4 sec) [Can just put the steps up one at a time but fairly quickly – this step is prereq knowledge]
Finally, in steps 1 and 2, we could instead have written this [Rcos(xα)] and used this identity… [cos(xα)cos(x)cos(α)sin(x)sin(α)]
But the rest of the process remains the same.
And actually, if you need to use this method in the exam, most of the time, they’ll tell you whether they want a sine function [emphasise Rsin(xα)] or a cos function. [emphasise Rcos(xα)]
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