Question

How do you express `cos(pi/ 3 ) * sin( ( 3 pi) / 8 )` without using products of trigonometric functions?

Answer 1

`cos (pi/3)*sin((3pi)/8)=1/2*sin ((17pi)/24)+1/2*sin (pi/24)`

Explanation:

start with `color(red)("Sum and Difference formulas")`

`sin (x+y)=sin x cos y + cos x sin y" " " "`1st equation
`sin (x-y)=sin x cos y - cos x sin y" " " "`2nd equation

Subtract 2nd from the 1st equation

`sin (x+y)-sin (x-y)=2cos x sin y`
`2cos x sin y=sin (x+y)-sin (x-y)`

`cos x sin y =1/2 sin (x+y)-1/2 sin (x-y)`

At this point let `x=pi/3` and `y=(3pi)/8`

then use

`cos x sin y =1/2 sin (x+y)-1/2 sin (x-y)`

`cos (pi/3)*sin((3pi)/8)=1/2*sin ((17pi)/24)+1/2*sin (pi/24)`

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