Question

How do you use the quadratic formula to solve `8cos^2theta-7costheta+1=0` for `0<=theta<360`?

Answer 1

`"see explanation"`

Explanation:

`"let " x=costheta`

`rArr8cos^2theta-7costheta+1=0`

`-=8x^2-7x+1=0`

`"solve using the "color(blue)"quadratic formula"`

`"with " a=8,b=-7,c=1`

`rArrx=(7+-sqrt(49-32))/16=7/16+-1/16sqrt17`

`rArrcostheta=7/16+1/16sqrt17" or " costheta=7/16-1/16sqrt17`

`rArrcostheta=0.6952" or " costheta=0.1798`

`rArrtheta=45.96^@" or " theta=79.64^@`

`"these are the "color(blue)"related acute angles"`

`"since " theta>0" in both cases then "theta" lies in"`
`"quadrants 1 and 4"`

`theta=45.96" or "theta=(360-45.96)`

`rArrtheta=45.96" or "314.04`

`color(red)"Also"`

`theta=79.64" or "theta=(360-79.64)`

`rArrtheta=79.64" or " 280.36`

`color(blue)"the solutions are"`

`theta=45.96^@,79.64^@,280.36^@,314.04^@to[0,360)`