Question

What is the value of sin(10)cos(20)sin(30)cos(40)sin(50)cos(60)sin(70)cos(80)? Not in radians, in degrees.

Thank you for your answer. No calculators.

Answer 1

Image reference...

Explanation:

The formulae I used,

`color(blue)(2sinacosa=sin2a)`
`color(red)(sin(90^@-theta)=costheta`
`color(red)(cos(90^@-theta)=sintheta`
`color(green)(sin(180^@-theta)=sintheta`

Just give a look....

For any further reference, please notify me.

Hope it helps....
Thank you...

Answer 2

`sin(10)cos(20)sin(30)cos(40)sin(50)cos(60)sin(70)cos(80)`

`=sin(90-80)cos(20)xx1/2xxcos(40)sin(90-40)xx1/2xxsin(90-20)cos(80)`

`=1/4xxcos^2(20)cos^2(40)cos^2(80)`

`=(1/2cos20cos40cos80)^2`

Now let `x=cos20cos40cos80`

`=>8xsin20=4*2sin20cos20cos40cos80`

`=>8xsin20=2*2sin40cos40cos80`

`=>8xsin20=2sin80cos80`

`=>8xsin20=sin160=sin(180-20)=sin20`

`=>x=1/8`

`=>cos20cos40cos80=1/8`

Hence given expression

`=(1/2cos20cos40cos80)^2`

`=(1/2*1/8)^2=1/256`