determine all co terminal angles that lie in the interval -4pi less thank or equal to theta less than or equal to 4pi, for the following angles.

a) 3pi/2
b) -5pi/3

To determine all the coterminal angles that fall in the given interval, we'll add or subtract multiples of 2π until we reach the desired interval. Let's solve each case:

a) For the angle 3π/2:
Adding 2π:
3π/2 + 2π = 7π/2, which is greater than 4π.
Subtracting 2π:
3π/2 - 2π = -π/2, which falls in the given interval.

Therefore, the coterminal angle that lies in the interval -4π ≤ θ ≤ 4π for 3π/2 is -π/2.

b) For the angle -5π/3:
Adding 2π:
-5π/3 + 2π = -5π/3 + 6π/3 = π/3, which falls in the given interval.

Subtracting 2π:
-5π/3 - 2π = -5π/3 - 6π/3 = -11π/3, which is less than -4π.

Therefore, the coterminal angle that lies in the interval -4π ≤ θ ≤ 4π for -5π/3 is π/3.

In summary:
a) The coterminal angle in the given interval for 3π/2 is -π/2.
b) The coterminal angle in the given interval for -5π/3 is π/3.

To determine the coterminal angles, we need to find angles that have the same terminal side as the given angle within the given interval.

a) Given angle: 3π/2

To find all the coterminal angles for 3π/2 within the interval -4π ≤ θ ≤ 4π, we can add or subtract any multiple of 2π to the given angle.

Adding 2π to 3π/2:
3π/2 + 2π = 7π/2

Subtracting 2π from 3π/2:
3π/2 - 2π = -π/2

So, the coterminal angles for 3π/2 within the given interval are 7π/2 and -π/2.

b) Given angle: -5π/3

To find all the coterminal angles for -5π/3 within the interval -4π ≤ θ ≤ 4π, we can add or subtract any multiple of 2π to the given angle.

Adding 2π to -5π/3:
-5π/3 + 2π = -5π/3 + (6π/3) = π/3

Subtracting 2π from -5π/3:
-5π/3 - 2π = -5π/3 - (6π/3) = -11π/3

So, the coterminal angles for -5π/3 within the given interval are π/3 and -11π/3.