How do you use the sum to product formulas to write the sum or difference $sin3theta+sintheta$ as a product?

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Answer 1 · 2017-09-23T09:04:56

$2sin2thetacostheta=4sinthetacos^2theta$

The relevant sum to product formula

$color(blue)(sinX+sinY=2sin((X+Y)/2)cos((X-Y)/2))$

we have

$sin3theta+sintheta$

$X=3theta,Y=theta$

so

$sin3theta+sintheta=2sin((3theta+theta)/2)cos((3theta-theta)/2)$

$=2sin2thetacostheta$

if you required it as single angles

using

$color(red)(sin2theta=2sinthetacostheta)$

$=4sinthetacos^2theta$

How do you use the sum to product formulas to write the sum or difference sin3theta+sinthetasin3theta+sintheta as a product?