Question

17. Solve each of the following equations for 0 (a) 4 cos 2x + 2 sin x = 3

(b) sin 2x = sin2x
(c) 7 sin x cos x = 2

Answer 1

a) A is a hard one but here goes...

`4cos2x + 2sinx = 3\\4cos2x + 2sinx -3 = 0\\4(1-sin^2x) +2 sinx - 3 = 0\\let w = sinx\\4(1-2w^2) + 2w -3 = 0\\8w^2 + 2w - 3 = 0\\factor\\\\(2w-1)(4w+3)=0\\\\w = sinx = (1)/(2),(-3)/(4\\)\\x= arcsin((1)/(2)) = (\pi)/(6) + 2k\pi \\\\and \\x =arcsin((-3)/(4) )+2k\pi`

b)

`sin2x = sin2x\\sin2x - sin2x = 0`

x = all real numbers

c)

`7sinx*cosx =2\\7 * (sin2x)/(2) = 2\\7*sin2x = 4\\sin2x = (4)/(7)\\2x = arcsin(4/7)\\x = (arcsin(4)/(7) )/(2)+2\pi k` where k is an integer.