Question

Question #91cd0

Answer 1

`theta = (7pi)/4+k * 2picolor(white)("XX")orcolor(white)("XX")theta =(5pi)/4+ k * 2pi`
`color(white)("XXXXXX") AAk in ZZ`

Explanation:

Given
`color(white)("XXX")sin(theta)+sqrt(2)=-sin(theta)`

`rarrcolor(white)("XXX")2sin(theta) = -sqrt(2)`

`rarrcolor(white)("XXX")sin(theta) = -sqrt(2)/2`

For `theta in [-pi/2,+pi/2]`
`rarrcolor(white)("XXX")theta=-pi/4 (or -45^@)`

For `theta in [0,2pi)`
`rarrcolor(white)("XXX")theta = (7pi)/4 or (5pi)/4`

For `theta` unrestricted, we need to allow for integer multiples of full circles:
`color(white)("XXX")theta = (7pi)/4+ k * 2pi or (5pi)/4 + k * 2pi`
`color(white)("XXXXXXXXX")`for any integer value `k`