Question

How do you find the solution to `5sin^2theta+2sintheta=0` if `0<=theta<2pi`?

Answer 1

`theta in {0,2.730,3.146,5.872}` (to 3 decimal places)

Explanation:

Given
`color(white)("XXX")5sin^2(theta)+2sin(theta)=0`

Factoring
`color(white)("XXX")sin(theta) * (5sin(theta)+2)=0`

`rArr` (for `theta in (-pi/2,pi/2)`)
#color(white)("XXX"){:(sin(theta)=0,color(white)("XX"),5sin(theta)+2=0), (rarr theta=0,,rarr sin(theta)=-2/5), (,,theta=-0.411516846 "( using calculator)") :}#

Since `sin(theta)=sin(theta+kpi), k in ZZ`
for the specified domain, `theta in [0,2pi)`
#color(white)("XXX"){: (theta= 0 or pi," or ",theta=pi-0.411516846 or 2pi-411516846), (,,theta=2.7300758075 or 5.8716684611) :}#