Question

A graph of y = cos(1/2 x) - sin( x) for -4ð x 4ð is shown in the figure. Assume z = 4.
(the figure is just the graph)

its asking me to find the x intercepts but i don't know how.

it also asks me to find: The x-coordinates of the eight turning points on the graph are solutions of sin(1/2x) + 2 cos(x) = 0. Approximate these x-coordinates to two decimal places.

(i need 8 answers for both)

please help!!!!

Answer 1

those weird signs next to the -4 and 4 mean the answer has to be between -4 pi and 4 pi

Answer 2

let's first change it to the same period.
sinx = 2sin(x/2)cos(x/2)

so you have
y = cos(x/2) - 2sin(x/2)cos(x/2)
for x=intercep, y = 0
cos(x/2)(1 - 2sin(x/2) = 0
cos(x/2) = 0 or sin(x/2) = 1/2
x/2 = π/2 or 3π/2 or x/2 = π/6 or 5π/6
x = π or 3π or π/3 or 5π/3

the period for sin(x/2) as well as cos(x/2) is 2π/(1/2) = 4π

so adding for subtracting 4π to any of my answers will produce a new answer, but of course we are only supposed to go as far as 4π, so there is no point adding 4π
let's subtract 4π
π -4π = -3π
3π-4π = -π
π/3 - 4π = -11π/3
5π/3 - 4π = -7π/3
so x = π or 3π or π/3 or 5π/3 or -3π or -π or -7π/3 or -11π/3


solving for the derivative equation
sin(x/2) +2cos(x) = 0
let's change cosx = 1 - 2sin^2 (x/2)

then sin(x/2) + 2(1 - 2sin^2(x/2)) = 0
4 sin^2 (x/2) - sin(x/2) - 2 = 0

let sinx = m and our equation becomes
4m^2 - m - 2 = 0
m = (1 ± √33)/8 = .84307 or -.59307

then sinx = .84307 or sinx = -.59307
use your calculator in radian mode, take sine inverse of each of those.
Each will result in two answers following the CAST rule.
good luck.