Question

How do you find the general solutions for `2sin^2 β – 3sin β +2 = 0`?

Answer 1

`2sin^2(beta)-3sin(beta)+2 = 0`
has no solutions

Explanation:

Consider the equation
`color(white)("XXX")2x^2-3x+2=0`
`color(white)("XXXXXX")`where `x` is (temporarilly) used in place of `sin(beta)`

The determinant of `2x^2-3x+2`
is `Delta =(-3)^2-4(2)(2) = -7`
and we know
`color(white)("XXX")` if `Delta < 0` the quadratic has no Real roots.

If there are no Real solutions for `x`
`rArr` there are no Real solutions for `sin(beta)`
`rArr` there are no solutions for `beta` (since `sin` is a Real valued function).

We could also see this by considering the graph of
`2sin^2(beta)-3sin(beta)+2`
graph{2(sin(x))^2-3sin(x)+2 [-10, 10, -4.08, 5.92]}
Note that this expression does not touch the X-axis (i.e. it is never equal to `0`)