What is the solution? ?
2sin^2 x +sinx +1=0

Call sin x = t.
#2t^2 - t - 1 = 0.#
Since a + b + c = 0. use shortcut. 2 real roots: t = 1 and #t = c/a = -1/2#
sin x = t = 1 --> #x = pi/2#
sin x = t = -1/2 --> #x = (7pi)/6# and #x = (11pi)/6#

Within interval #(0, 2pi)#: 3 answers: #pi/2; (7pi)/6 , and (11pi)/6.#

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step-by-step explanation:

d = distance between moon and sun, which in this problem is the hypotenuse of a right triangle.  

the distance y is the side opposite to angle x in the right triangle.   use the definition of the sine function from basic trigonometry.

sin(x) = y/d ⇒

d = y/sin(x)

c

step-by-step explanation: