Question

Solve the equation `sectheta+costheta=5/3`?

Answer 1

`sec theta -cos theta = +-sqrt(-11/9) = +-sqrt(11/9) i`

Explanation:

`sec theta +cos theta =5/3 :. (sec theta +cos theta)^2 =25/9` or

`sec^2 theta +cos^2 theta + 2 sec theta*cos theta=25/9` or

`sec^2 theta +cos^2 theta + 2 cancelsec theta*1/cancelsec theta=25/9` or

`sec^2 theta +cos^2 theta =25/9 -2=7/9`

`(sec theta -cos theta)^2 = sec^2 theta +cos^2 theta - 2 sec theta*cos theta = 7/9-2 =-11/9`

`sec theta -cos theta = +-sqrt(-11/9) = +-sqrt(11/9) i` [Ans]

Answer 2

We cannot have `sectheta+costheta=5/3 < 4`

Explanation:

As `(sectheta+costheta)^2=(sec theta-costheta)^2+4secthetacostheta`

= `(sec theta-costheta)^2+4`

As `(sec theta-costheta)^2 >=0`,

we have `(sec theta+costheta)^2 >= 4`

and hence `sec theta+costheta >=2`

or `sectheta+costheta <= -2`

This is also seen from the graph of `secx+cosx`
graph{secx+cosx [-10, 10, -5, 5]}

Hence, we cannot have `sectheta+costheta=5/3 < 4`

Also observe that `sectheta-costheta` can take all values.

graph{secx-cosx [-10, 10, -5, 5]}