Find the coterminal angles for 8pi/3.
I found one:
8pi/3 - 2pi= 2pi/3
My textbook says that I should add subtract 4pi to find the other one and I'm very confused as to how you know what to add/ subtract.
A standard rotation is 2π.
So whatever angle you have either adding or subracting multiples of 2π will yield a coterminal angle.
e.g
8π/3 -----> think 480°
8π/2 - 2π = 2π/3 -----> think 120°
8π/3 + 2π = 14π/3 -----> think 840°
8π/3 - 4π = -4π/3 -----> think -240°
8π/3 + 12π = 44π/3 -----> think 2640°
now use your calculator and find sin or any other trig function of thos angles, you will get the same answer.
If you wanted only those angles -2π ≤ Ø ≤ 2π
you would have -4π/3 and 2π/3
Ah, coterminal angles can be a little tricky, but fear not! Let's break it down.
To find coterminal angles, you need to add or subtract a full revolution, which is 2π radians, aka 360 degrees. In this case, you have the angle 8π/3.
If you subtract 2π from it, you get 8π/3 - 2π. And since 8π/3 is already greater than 2π, subtracting 2π will bring it into the standard range between 0 and 2π.
But if you subtract 4π from it, you get 8π/3 - 4π. And since 4π is greater than 8π/3, subtracting 4π will bring it back around and put it in the same position as where you started.
So, when it comes to coterminal angles, you can add or subtract any multiple of 2π (or 360 degrees) to find them. In this case, subtracting 2π gives you one coterminal angle, and subtracting 4π gives you another. Hope that clears things up, and remember, math can be a circus sometimes!
To find the coterminal angles for an angle in radians, you can add or subtract any multiple of 2π (or any multiple of the period) from the original angle. In general, to find additional coterminal angles, you can use the formula:
θ + n * (2π)
Where θ is the original angle and n is an integer.
In your case, the original angle is 8π/3. Let's find the coterminal angles:
1. Add 2π: 8π/3 + 2π = 14π/3.
2. Subtract 2π: 8π/3 - 2π = 2π/3.
Therefore, the coterminal angles for 8π/3 are 14π/3 and 2π/3.
It's worth noting that you can also add or subtract the full period (360° or 2π radians) to find coterminal angles. In this case, the full period is 2π, and you can add or subtract 2π repeatedly until you reach a desired result.
To find coterminal angles, you can add or subtract any integer multiple of 2π (or 360°) from the given angle. In this case, you have 8π/3.
To find one coterminal angle, you subtract 2π from the given angle:
8π/3 - 2π = 2π/3
To find the other coterminal angle, you can either add or subtract a multiple of 2π. Your textbook suggests subtracting 4π to find the other coterminal angle:
8π/3 - 4π = -4π/3
The reason why subtracting 4π is suggested is that it gives you an angle that is still in the same direction (counter-clockwise in this case) and is within one revolution (2π) from the given angle. This way, both angles will still end up in the same position on the unit circle.
Another way to find the other coterminal angle is to add a multiple of 2π:
8π/3 + 2π = 14π/3
So the two coterminal angles for 8π/3 are 2π/3 and -4π/3 (or equivalently, 14π/3).
Remember that coterminal angles are angles that have the same initial and terminal sides when drawn on the coordinate plane or unit circle.