Question

Find the exact value? `3-2sin^2x=3cosx`

Answer 1

`rarrx=2npi+-pi/3` OR `x=2npi`

Explanation:

`rarr3-2sin^2x=3cosx`

`rarr3-2(1-cos^2x)=3cosx`

`rarr3-2+2cos^2x=3cosx`

`rarr2cos^2x-3cosx+1=0`

`rarr2cos^2x-2cosx-cosx+1=0`

`rarr2cosx(cosx-1)-1(cosx-1)=0`

`rarr(2cosx-1)(cosx-1)=0`

Either, `2cosx-1=0`

`rarrcosx=1/2=cos(pi/3)`

`rarrx=2npi+-pi/3` where `nrarrZ`

OR, `cosx-1=0`

`rarrcosx=1=cos 0`

`rarrx=2pi n` where `nrarrZ`

Answer 2

`x = pm 60^circ + 360^circ k or x = 360^circ k quad` integer `k`

Explanation:

The other answer is fine; I just want to add this to my collection of yet anther trig problem that's going to work out to 30/60/90 or 45/45/90. When are we going to realize that it's goofy to teach a subject that only handles two triangles reasonably well?

`3 - 2 sin ^2 x = 3 cos x`

`3 - 2(1 - cos^2 x) =3 cos x`

`2 cos ^2x - 3 cos x + 1 = 0`

`(2 cos x - 1 )(cos x - 1) = 0`

`cos x = 1/2 or cos x = 1`

Cosine or sine of `1/2`; same old same old.

`cos x = cos 60^circ or cos x = cos 0`

`x = pm 60^circ + 360^circ k or x = 360^circ k quad` integer `k`