Refrence Angles & Positive/Negative Angles(Coterminals) HOW?

[mathzeroh]

my question is, how do you "Find one positive angle and one negative angle that are coterminal with each angle." :

For example:

$$\frac{5\pi}{12}$$


And also, I had a question on this as well:

Find the reference angle for each angle with the given measure.
$$\frac{12\pi}{5}$$

i noe that you have to see what quadrant it is in, then from there, you use the Reference Angle Rule, whichever one. But what if the measure is a negative number? Like for example, $$-210\circ$$?? What would you do here?

if anyone can help me with these i'd be really gratefull! thanks!

 

[AKG]

my question is, how do you "Find one positive angle and one negative angle that are coterminal with each angle."

Given some angle, x, you can find a positive angle co-terminal with x by adding $2\pi$ to x until you get a positive number, and you can figure out how to find a negative angle.

Find the reference angle for each angle with the given measure.
$$\frac{12\pi}{5}$$

To find the reference angle for any angle, keep continuing to add or subtract $2\pi$ until you get a value, x, such that $0 \leq x < 2\pi$. Of course, you'll have to figure out whether you need to add, subtract, or neither.

 

[M98Ranger]

To find a positive and negative coterminal angle with a given angle, you can add or subtract 2π (or 360°) from the given angle. This is because 2π (or 360°) is the period of a circle, meaning that adding or subtracting it will bring you back to the same angle.

For example, if the given angle is \frac{5\pi}{12}, you can add 2π to get a positive coterminal angle: \frac{5\pi}{12} + 2\pi = \frac{29\pi}{12}. You can also subtract 2π to get a negative coterminal angle: \frac{5\pi}{12} - 2\pi = -\frac{19\pi}{12}.

For the angle \frac{12\pi}{5}, you can use the same method. Adding 2π gives a positive coterminal angle: \frac{12\pi}{5} + 2\pi = \frac{22\pi}{5}. Subtracting 2π gives a negative coterminal angle: \frac{12\pi}{5} - 2\pi = -\frac{8\pi}{5}.

When finding the reference angle for a given angle, you need to consider the quadrant in which the angle lies. For positive angles, the reference angle will always be the angle formed between the terminal side of the given angle and the x-axis. For negative angles, the reference angle will be the angle formed between the terminal side of the given angle and the negative x-axis.

For the angle -210°, you can first convert it to radians by multiplying by \frac{\pi}{180}. This gives -\frac{7\pi}{6}. Since this angle falls in the third quadrant, the reference angle will be the angle formed between the terminal side and the negative x-axis, which is \frac{\pi}{6}.

I hope this helps with your understanding of reference angles and coterminal angles. Remember to always consider the quadrant and use the period of 2π (or 360°) when finding coterminal angles.