Use the substitution method and then factor
Let u=sinxu=sinx
so, u^2=sin^2xu2=sin2x
2color(red)(u^2)-5color(red)(u)+2=02u2−5u+2=0
Now you can factor
Multiply the coefficient of the first term, 22, with the last term, 22.
2*2=42⋅2=4
Ask yourself what are the factors of 44 that add up to the coefficient of the middle term, -5−5?
Factors of 4
(1)(4)(1)(4) => NO⇒NO
color(red)((-1)(-4))(−1)(−4) => color(red)(YES)⇒YES
(2)(2)(2)(2) => NO⇒NO
(-2)(-2)(−2)(−2) => NO⇒NO
Now place those factors in an order that makes it easy to factor by grouping.
(2u^2color(red)(-4u))+(color(red)(-1u)+2)=0(2u2−4u)+(−1u+2)=0
Factor out 2u2u from the first grouping.
Factor out -1−1 from the second grouping.
color(red)(2u)(u-2)color(red)(-1)(u-2)=02u(u−2)−1(u−2)=0
Now you can factor out a grouping, (u-2)(u−2)
(u-2)(2u-1)=0(u−2)(2u−1)=0
Now use the Zero Property
u-2=0u−2=0 and 2u-1=02u−1=0
u=2u=2 and u=1/2u=12
Now switch back to sinxsinx
sin x=2sinx=2 and sin x =1/2sinx=12
cancel(sin x=2) is discarded because sin x oscillates between -1 and 1.
Falling back to trigonometry sin x is in the ratio y/r
y/r=1/2 which means that x=sqrt3 by using the pythagorean theorem x=sqrt(2^2-1^2)=sqrt(4-1)=sqrt3 or knowing that we have the special triangle , 30,60,90 which corresponds to the sides 1,sqrt3,2.
The side of length 1 corresponds to 30 degrees which is also pi/6
All of that to say that
sin (pi/6)=1/2
x={pi/6}
I have tutorials on methods of factoring found here, https://www.youtube.com/playlist?list=PLsX0tNIJwRTxqIIcfpsTII9_xhqVIbQlp
