I thought this would be the answer but my answer key says -1/2 how is this determined. What am I doing wrong?

SOH CAH TOA

Thanks for the NSFW tag. I almost opened this in front of my kids!
What picture did you draw to relate this to 4pi/3?

The angle 4pi/3 is in quadrant 3. Both the x and y coordinates (cosine and sine) are negative. If you draw the picture right, you would see that the relevant triangle has adjacent 1/2, but it’s to the left of the origin so it’s an x coordinate of -1/2.
Using triangles for angles > pi is so convoluted. You can’t draw a triangle with angles > pi, and if you minus pi you end up flipping sign. Wouldn’t it be easier to draw the unit circle with radius 1? Or at least set the hypotenuse = 1 so that the sides already correspond to cos (adjacent) and sin (opposite), but then you still need to remember to change sign!
Well first of all your hypotenuse should be 1 since you're dealing with a unit circle.

Secondly, 4pi/3 is equal to 4/3 *pi. Which would indicate that it's 180° + 1/3 pi. Now just take the cosine of 1/3 pi and negate it since you're going into the negative-negative quadrant and there's your answer. (for simplicity, 1/3 pi is 60° so it's a 30/60/90 triangle)
Cos 2A= 2(Cos(A))^2   -  1

You can solve this question using bifurcation as below.

Cos(4pi/3)=2Cos^2(2pi/3)-1

Cos(2pi/3)=2Cos^2(pi/3)-1

We have Cos(pi/3)=1/2

Cos(2pi/3)=2(1/2)^2-1=-1/2

Cos(4pi/3)=2Cos^2(2pi/3)-1=2(-1/2)^2-1=(1/2)-1=-(1/2)

Hence proved.
Draw the trigonometric circle and mark the angle on it, the SOH CAH TOA definition only works for angles smaller than pi/2
Well, you could do a rule of three to find the angle 4π/3 in degrees. We know that π equals to 180°, so 4π/3 will be 240°.

From here on you have two simple ways I can think of. One: 240 = 180 + 60, so you could use the cos(a + b) identity.

Two: 240 = 180 + 60, so you could interpretate this as 60° after 180° degrees in the trigonometric circle. The trigonometric circle is basically a quarter of a circle that repeats itself 4 times. The value of the cossines and sines remain numeracally but change signals as it change quarters. We know that 180 is the beginning of a new quarter, so we can say that we are in the beginning of the quarter, the 0° position. So we are left with cossine(0° + 60°) after the beginning. As this quarter of the circle is negative for cossine values you would get -cossine(60)
Imagine the cos line, at 30° (1/3 pi) its 1/2. Cos between 0 and pi is same line as pi and 2 pi, just mirror image along the x axis.

If you shift the pi to 2pi fragment below the 0 to pi you will see that the distance from the x axis at 1/3 pi both of the original cos values and the mirror image below are the same. This holds true for any value of x. So answer is -1/2
Unit circle baby