we have that
[tex] 2sin^{2} x-5sin x-3=0 [/tex]
I. Rewrite the equation by substituting the expression u in for sin x.
[tex] 2u^{2} -5u-3=0 [/tex]
II. Factor the quadratic expression. Rewrite the equation with factors instead of the original polynomial.
[tex] 2u^{2} -5u-3=0 [/tex] is equal to
using a graph calculator-----> see the attached figure
[tex] (u-3)*(2u+1)=0 [/tex]
III. Use the zero product property to solve the quadratic equation.
[tex] (u-3)*(2u+1)=0 [/tex]
(u-3)=0--------------> u=3
(2u+1)=0-------- 2u=-1------> u=-1/2-----> u=-0.5
IV. Rewrite your solutions to Part III by replacing u with sin x.
sin x=3--------> is not the solution (sin x can not be greater than 1)
sin x=-0.50------>is the solution
V. Solve the remaining equations for x, giving all solutions to the equation.
sin x=-0.50
if the sine is negative
then
x belong to the III or IV quadrant
we know that
sin 30°=0.50
so
the solution for the III quadrant is
x=180°+30°-------> x=210°
the solution for the IV quadrant is
x=360°-30°------> x=330°
