Question

How do you solve `Sin 2 theta + cos theta = 0`?

Answer 1

Assuming an interval of `[0,2pi]`
`theta in {pi/2,(3pi)/2,(7pi)/6,(11pi)/6}`

Explanation:

Using the double angle formula:
`color(white)("XXX")sin(2theta)=2sin(theta)cos(theta)`

`sin(2theta)+cos(theta)=0`

`rArr 2sin(theta)cos(theta)+cos(theta)=0`

`rArr(cos(theta))*(2sin(theta)+1)=0`

`rArr` (within the interval `theta in [0,2pi]`)
`color(white)("XXX"){: (cos(theta)=0,color(white)("XXX")"or"color(white)("XXX"),2sin(theta)+1=0), (rarrcolor(white)("X")theta=pi/2" or " (3pi)/2,,rarrcolor(white)("X")sin(theta)=-1/2), (,,rarrcolor(white)("X")theta=(7pi)/6" or "(11pi)/6) :}`