Name: Allan

Who is asking: Teacher
Level: Secondary

Question:
solve:

sin 2x = cos 3x

Primary question: how do you handle the cos 3x?



Hi Allan,

You can use the multiple angle expressions for sin(x+y) and cos(x+y) to write cos(3x) in terms of cos(x) and sin(x).

cos3x = cos(2x + x)
= cos(2x)cos(x) - sin(2x)sin(x)
= [1 - 2sin2(x)]cos(x) - 2sin(x)cos(x)sin(x)
= cos(x) - 2sin2(x)cos(x) - 2sin2(x)cos(x)
= cos(x) - 4sin2(x)cos(x) Note that in line 3, a different formula could be used for cos(2x), but looking ahead you can see that this will work best for solving the equation, since sin(x)cos(x) terms will show up on both sides.

Also

sin2x=2sin(x)cos(x)

Equating the expressions for cos(3x) and sin(2x) you can extract a factor of cos(x) and be left with a quadratic in sin(x) that can be solved using the general quadratic.

Paul and Chris
Go to Math Central