Double Angle and Half Angle Formulae: Summary

Here’s a reminder of the key points you should know about the double and half angle identities.

A double angle is any compound angle that is the sum of two equal angles.

There are double angle identities for sin, cos and tan which allow us to rewrite trig functions without the double angles. [show them greyed out]

First, the double angle identity for sine is this. [sin(2A) 2sin(A)cos(A)] [zoom in on it or grey out/ remove the other two] 

If an equation we’re trying to solve contains sine functions with different multiples of the same input…

… this identity may provide a route to solving the equation.(wait 3 sec) [3sin (2x) -2sin x =0  →  3(2sin(x)cos(x)) – 2sin(x) =0 keep the identity on screen here and make it clear the green bit is what’s changing. Then show the next few lines of working too. 6sin(x)cos(x) -2sin(x) = 0 → 2sin(x)(3cos(x)-1)=0 → 2sin(x)=0 or 3cos(x)=1.] 

Or, if we have sine and cosine functions multiplied together, we can use this identity in reverse. [e.g. 8sin(x)cos(x) =3 → 4sin(2x) =3 keep the identity on screen here and make it clear the green bit is what’s changing] (wait 2 sec)

Second, the double angle identity for cosine can be written in 3 ways. [cos(2A) cos2(A)-sin2(A) or cos(2A)1-2sin2(A) or cos(2A)2cos2(A)-1 but can grey out “cos(2A)” for the 2nd and 3rd ones]

If an equation we’re trying to solve contains cos functions with different multiples of the same input… [cos(2x) = 3cos(x) -4]

Any of these identities may provide a route to solving the equation. [Show using the bottom of the 3 identities to change to 2cos²(x)-1 = 3cos(x) -4…. Then 2cos²(x)-3cos(x)+3=0]

And it doesn’t matter which of the three we use: if one of them works, all three of them will work eventually!

But this flowchart can gives us a good idea of which identity is most likely to be useful.(wait 5 sec) [This but include the full identity in each box.]

Alternatively, if we recognise that the equation we want to solve contains a term in one of these forms…(click once) [show all the identities and highlight the RHSs]

Third, the double angle identity for tan is this. [tan(2A) 2tan(A)1-tan2(A)]

If an equation we’re trying to solve contains tan functions with different multiples of the same input…

This identity may provide a route to solving the equation.

[Show this example making it clear that we’re using the identity to change the green bit: 6tan (A) =  tan (2A)  →6tan (A)= 2tan(A)1-tan2(A) ]

… or when part of the equation is in this form, we can use the identity in reverse. [tan(A)1-tan2(A)  And then show this example making it clear that we’re using the identity to change the green bit 4tan(x)1-tan2(x)=3  → 2tan(2A)=3]

And all of the double angle formulae can work with other angles too…

… as long as these angles are always double these angles. [Show sin(2A)2sin(A)cos(A), cos(2A)cos2(A)-sin2(A), tan(2A)2tan(A)1-tan2(A) and highlight all the 2A’s at once followed by all copies of just A at once (not the 2A’s this time).]

So, for instance, these all hold true. (wat for animation and wait 3 sec) [For these 3 examples, the important thing is that the angle on the left is twice the angle on the right so let’s highlight the angles.] [Maybe change one example at a time so they can focus on one at a time. So first for the sin(2A) example, show the equation, highlight 2A and A, then change to 20 and 10. Then same for the other examples.]

[sin(2A)2sin(A)cos(A) → sin(20)2sin(10)cos(10)]

[cos(2A)cos2(A)-sin2(A) → cos(4)cos2(A)-sin2(2)]

[tan(2A)2tan(A)1-tan2(A) → tan(25)2tan(12.5)1-tan2(12.5)]

Finally, here are the half angle identities for sine… [sin(x2)1-cos(x)2]

… and cos. [cos(x2)1+cos(x)2]

And you might find it useful to remember these, but technically the double angle identities can do anything these can, so memorising these is not strictly necessary.