Some of your work might not show up, since the symbols were pictures and not text.
Determine the general solution to sin x/2 = 3/ 2. Use radian measure in your solution.
The solution to the equation sin x/2 = 3/ 2 is given by intersection points, of both graphs. Points of intersection occur at x=120° and 239° when domain is 0°< x< 360°. Since the period is 360°, the general solutions are:
N= any integer x= 120° + n (360°) and x=239°+ n (360°)
Which then would together form the equation x= (360°) n+ 120°
I would have done this using the half angle formula(did you learn that?):
sin(x/2) = sqrt[(1 - cosx)/2]
=> sqrt[(1 - cosx)/2] = sqrt(3)/2
=> (1 - cosx)/2 = ¾
=> 1 - cosx = 3/2
=> cosx = 1 – 3/2 = -1/2
=> x = cos^-1(-1/2)
=> General Solution: x = 2pi/3 + 2kpi, x = 4pi/3 + 2kpi, for k some integer.
Since I squared the equation, let's see if any solution is extraneous:
=> sin(pi/3) = sqrt(3)/2 ........Check!
=> sin(2pi/3) = sqrt(3)/2 ......Check, both answers check out.
So your answers should be in radians, as mine are. Also, it's 240 degrees not 239.
Doing it your way you'd have to be familiar with the sine graph to know that sin(theta) = sqrt(3)/2 when theta = pi/3 and theta = 2pi/3
so x/2 = pi/3 => x = 2pi/3
or x/2 = 2pi/3 => x = 4pi/3
and we add 2kpi to each of these for the general solution.
2) Solve for x, where 0<x<2.
a. 2 tan x + 12 = 0
2 tan x + 12 = 0
2 tan x/2 + 12/2 = 0
Tan x= - 3
X= 2/3, 5/ 3
I wouldn't have done this much different. Well, not really, I would never have switched to tan(x/2), there's no need for a half angle.
2tanx + sqrt(12) = 0
=> 2tanx = - sqrt(12)
=> tanx = -sqrt(12)/2
=> tanx = -sqrt(4)sqrt(3)/2
=> tanx = - sqrt(3)
=> x = tan^-1(-sqrt(3))
=> x = 2pi/3, 5pi/3
b. 2x + cos x=1
2cos^2x + cos x=1
2cos^2x + cos x-1=0
x= /3,, 5/ 3,
You factorized incorrectly:
2cos^2x + cosx – 1 = 0
=> (2cosx – 1)(cosx + 1) = 0
=> 2cosx – 1 = 0
=> cosx = ½
=> x = pi/3 + 2kpi, x = 5pi/3 + 2kpi, for k some integer .........general solution
cosx + 1 = 0
=> cosx = -1
=> x = pi + 2kpi, for k any integer .........general solution
So x= pi + 2kpi, pi/3 + 2kpi, x = 5pi/3 + 2kpi, for k any integer .........general solutions
Write the general solutions to the equation 2.a. and 2.b.
2a) 2/3+ n,5/ 3+ n are the two general solutions
2B) 5/ 3+2n, n= any integer
2a) is fine, 2b) is incorrect, see the problem above