Math Snap

STEP 1

What is this asking?
We need to find one positive and one negative angle that are coterminal with $-\frac{29\pi}{5}$ radians, meaning they point in the same direction, but have different rotations.
Watch out!
Remember, coterminal angles share the same initial and terminal sides, but can be reached by rotating different amounts.
Don't forget to simplify your answers!

STEP 2

1. Find a positive coterminal angle.
2. Find a negative coterminal angle.

STEP 3

Alright, let's start with our given angle, $-\frac{29\pi}{5}$.
Since it's negative, we'll need to add a whole number of full rotations to make it positive.
A full rotation is $2\pi$ radians.

STEP 4

Let's add $2\pi$ to our angle:
$$-\frac{29\pi}{5} + 2\pi$$ To add these, we need a common denominator.
We can rewrite $2\pi$ as $\frac{10\pi}{5}$.
Now we have:
$$-\frac{29\pi}{5} + \frac{10\pi}{5} = -\frac{19\pi}{5}$$ This is still negative!
We need to keep adding full rotations.

STEP 5

Let's add another $2\pi$, which is the same as adding another $\frac{10\pi}{5}$:
$$-\frac{19\pi}{5} + \frac{10\pi}{5} = -\frac{9\pi}{5}$$ Still negative!
One more rotation should do it!

STEP 6

Adding another $\frac{10\pi}{5}$:
$$-\frac{9\pi}{5} + \frac{10\pi}{5} = \frac{\pi}{5}$$ Woohoo! This is a positive angle coterminal with our original angle.

STEP 7

Now, let's find a negative coterminal angle.
We can do this by subtracting $2\pi$ from our original angle, $-\frac{29\pi}{5}$.

STEP 8

Remember, $2\pi$ is the same as $\frac{10\pi}{5}$, so we have:
$$-\frac{29\pi}{5} - \frac{10\pi}{5} = -\frac{39\pi}{5}$$ This is negative and distinct from our original angle, so we've found our negative coterminal angle!

Positive coterminal angle: $\frac{\pi}{5}$ radians
Negative coterminal angle: $-\frac{39\pi}{5}$ radians