How do you express sin(π6)cos(3π4) without using products of trigonometric functions?

1 Answer

Using Sum or Difference
12sin(11π12)12sin(7π12)

Explanation:

From "Sum and Difference Formulas"

sin(x+y)=sinxcosy+cosxsiny
sin(xy)=sinxcosycosxsiny

Add the equations to obtain

sin(x+y)+sin(xy)=2sinxcosy

so that, after dividing both sides by 2

sinxcosy=12sin(x+y)+12sin(xy)

at this point , Let x=π6 and y=3π4

sinxcosy=12sin(x+y)+12sin(xy)

sin(π6)cos(3π4)=12sin(π6+3π4)+12sin(π63π4)

sin(π6)cos(3π4)=12sin(11π12)+12sin(7π12)

but sin(7π12)=sin(7π12)

therefore

sin(π6)cos(3π4)=12sin(11π12)12sin(7π12)

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