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!!!!

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

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.

To find the x-intercepts of the graph y = cos(1/2x) - sin(x), we need to find the values of x where y is equal to zero. In other words, we are looking for the x-values where the graph crosses the x-axis.

Here's how you can find the x-intercepts:

1. Set the equation y = cos(1/2x) - sin(x) equal to zero: 0 = cos(1/2x) - sin(x).

2. Rearrange the equation to isolate the trigonometric terms: sin(x) = cos(1/2x).

3. Take the square of both sides of the equation to eliminate the trigonometric functions so that you have a quadratic equation: sin^2(x) = cos^2(1/2x).

4. Apply the double-angle formula for cosine: 1 - sin^2(x) = cos^2(1/2x).

5. Replace sin^2(x) with 1 - cos^2(x) using the Pythagorean identity: 1 - (1 - cos^2(x)) = cos^2(1/2x).

6. Simplify the equation: 2cos^2(x) = cos^2(1/2x).

7. Bring all the terms to one side: 2cos^2(x) - cos^2(1/2x) = 0.

8. Factor out a common factor of cos^2(x): cos^2(x) (2 - cos(1/2x)) = 0.

Now you have two equations:

a) cos^2(x) = 0.
b) 2 - cos(1/2x) = 0.

For equation a), cos^2(x) = 0 has a solution of x = ±π/2, which corresponds to the x-intercepts of the graph.

For equation b), 2 - cos(1/2x) = 0, you need to solve for x by isolating the cosine term:

1. Subtract 2 from both sides of the equation: cos(1/2x) = -2.

2. Since the cosine function has a maximum value of 1 and a minimum value of -1, there are no real solutions for the equation cos(1/2x) = -2. Therefore, there are no additional x-intercepts.

So, the x-intercepts of the graph y = cos(1/2x) - sin(x) are x = ±π/2.

Next, let's find the x-coordinates of the turning points on the graph that are solutions of sin(1/2x) + 2cos(x) = 0.

1. Set the equation sin(1/2x) + 2cos(x) = 0.

2. Rearrange the equation to isolate the trigonometric terms: sin(1/2x) = -2cos(x).

3. Square both sides of the equation to eliminate the trigonometric functions: sin^2(1/2x) = 4cos^2(x).

4. Apply the double-angle formula for sine: (1 - cos(x))/2 = 4cos^2(x).

5. Multiply both sides of the equation by 2 to eliminate the fraction: 1 - cos(x) = 8cos^2(x).

6. Rewrite the equation using the Pythagorean identity: 1 - cos^2(x) = 9cos^2(x).

7. Simplify the equation: 10cos^2(x) = 1.

8. Divide both sides of the equation by 10: cos^2(x) = 1/10.

Taking the square root of both sides, we have:

cos(x) = ±sqrt(1/10).

To find the x-coordinates, we need to find the angles x that correspond to the given cosine values.

Using a calculator, find the values of the inverse cosine function (cos^(-1)) for ±sqrt(1/10) to obtain the approximate values of x. Round each value to two decimal places.

For cos(x) = sqrt(1/10), x ≈ ±1.77, ±4.36, ±6.30, ±8.73. (Please note that these are approximate values rounded to two decimal places.)

For cos(x) = -sqrt(1/10), x ≈ ±2.97, ±5.76, ±7.38, ±9.13. (Again, these are approximate values rounded to two decimal places.)

These are the approximate x-coordinates of the eight turning points on the graph.