Solve the following. Find all solutions from 0 degrees to 360 degrees.
1. 4 sin^2 x – 1 = 0
2. sin^2 x + sin x = 1
1. 4 sin^2 x – 1 = 0 add 1 to both sides
4sin^2x = 1 divide both sides by 4
sin^2 x = 1/4 take both roots and we have that
sin x = √[1/2 ] and this happens at 45 degrees and 135 degrees
And
sin x = - √[1/2] and this happens at 225 degrees and 315 degrees
2. sin^2 x + sinx = 1
sin^2 x + sin x - 1 = 0
Let a = sin x
a^2 + a - 1 = 0
a^2 + a= 1 complete the square on a
a^2 + a + 1/4 = 1 + 1/4
(a + 1/2)^2 = 5/4
Take both roots
a + 1/2 = √5/2
a = √5 - 1
_____
2
sin x = √5 - 1
_____ take the arcsin and x ≈ 38.17° and (180 - 38.17) ≈ 141.8°
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Or
a = -√5 + 1
______
2
sin (x) = - √5 - 1
_______ this answer is < -1....so no answer for sin x
2