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For any two complex waves ​f(x)=r1​sin(ax+α1​)​ and ​g(x)=r2​sin(ax+α2​)​ with the same frequency but different phase, we can use Euler's form of complex numbers to express ​h(x)​ as a single sine function:
So we can write
where
Note: In most questions about adding complex waves, you can use your graphing calculator to find the phase shift instead of deriving it by hand.