Question

How do you solve `2sin^2x+7sinx=4` in the interval [0,360]?

Answer 1

`x in {30^@, 150^@, 194.48^@,345.52^@)`

Explanation:

Given
`color(white)("XXX")2sin^2(x)+7sin(x)=4`
Rearranging into standard quadratic form:
`color(white)("XXX")2sin^2(x)+7sin(x)-4=0`
Factoriing:
`color(white)("XXX")(2sin(x)-1)(sin(x)+4)=0`
`rarr`
#color(white)("XXX"){: (sin(x)=1/2," or ",sin(x)=-1/4), ("in the range",[0,360^@],), (x=30^@" or "150^@,,x~~194.48^@" or "345.52^@) :}#

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Note: `x` is one of the standard angles for `sin(x)=1/2`;
but I needed to use a calculator for `arcsin(-1/4)` to evaluate `sin(x)=-1/4`