Find Angles Coterminal With -60 Degrees

Hey math whizzes! Ever feel like angles are just repeating themselves? Well, you're kind of right! Today, we're diving into the awesome world of coterminal angles. These are angles in standard position that share the same terminal side. Think of it like spinning around on a merry-go-round – you can go around multiple times, forwards or backwards, and still end up in the same spot. That spot is your terminal side! So, if you're given an angle, say $-60^{\circ}$, any other angle that lands you on that exact same ray after rotating is considered coterminal. The key to finding these guys is understanding that a full circle is $360^{\circ}$. Adding or subtracting multiples of $360^{\circ}$ to your original angle will always result in a coterminal angle. It's like adding or subtracting full laps around that merry-go-round. We're going to tackle a problem where we need to identify all the coterminal angles from a given list that match our starting angle of $-60^{\circ}$. Get ready to do some adding and subtracting of $360^{\circ}$!

Understanding Coterminal Angles: The Big Picture

So, what exactly are coterminal angles, and why should you even care, guys? Imagine you're drawing angles on a coordinate plane. Standard position means the angle's vertex is at the origin, and its initial side lies along the positive x-axis. The terminal side is where the angle ends up after you rotate. Now, coterminal angles are those that, when drawn in standard position, have their terminal sides overlapping perfectly. This means they represent the same position on the circle, even though they might have been reached by different amounts of rotation. For example, $30^{\circ}$ and $390^{\circ}$ are coterminal because $30^{\circ} + 360^{\circ} = 390^{\circ}$. You could also have negative coterminal angles. If you go $-30^{\circ}$ (clockwise rotation), that's the same as going $330^{\circ}$ (counter-clockwise rotation), since $-30^{\circ} + 360^{\circ} = 330^{\circ}$. The general rule is: if $\theta$ is an angle, then $\theta + n \cdot 360^{\circ}$ is a coterminal angle for any integer $n$ (where $n$ can be positive, negative, or zero). This formula is your golden ticket to finding or verifying coterminal angles. In our specific problem, we're starting with $-60^{\circ}$. So, we're looking for angles that can be expressed as $-60^{\circ} + n \cdot 360^{\circ}$. This means we'll be testing each option by seeing if subtracting or adding multiples of $360^{\circ}$ from it results in $-60^{\circ}$. It's all about finding that common landing spot on the unit circle!

How to Find Coterminal Angles: Step-by-Step

Alright, let's get down to business on how to actually find these elusive coterminal angles. Our starting point is $-60^{\circ}$. Our mission, should we choose to accept it, is to sift through the given options and pick out the ones that are coterminal with $-60^{\circ}$. Remember our trusty formula: an angle $\alpha$ is coterminal with $\theta$ if $\alpha = \theta + n \cdot 360^{\circ}$ for some integer $n$. In our case, $\theta = -60^{\circ}$. So, we're looking for options $\alpha$ such that $\alpha = -60^{\circ} + n \cdot 360^{\circ}$. This can be rearranged to check if an angle is coterminal: $\alpha - (-60^{\circ}) = n \cdot 360^{\circ}$, which simplifies to $\alpha + 60^{\circ} = n \cdot 360^{\circ}$. This means that if you add $60^{\circ}$ to any coterminal angle, the result should be a multiple of $360^{\circ}$. This is the most straightforward way to check each option. Let's go through the list:

• $-1020^{\circ}$: Is $-1020^{\circ} + 60^{\circ}$ a multiple of $360^{\circ}$? $-1020^{\circ} + 60^{\circ} = -960^{\circ}$. Is $-960^{\circ}$ a multiple of $360^{\circ}$? Yes, $-960^{\circ} / 360^{\circ} = -2.666...$. Uh oh, that's not an integer! So, $-1020^{\circ}$ is not coterminal. Let's recheck that division. $-960 / 360 = -96/36 = -8/3$. Still not an integer. Hmm, maybe my initial thought was wrong. Let's try the other way: $-1020 = -60 + n * 360$. $-1020 + 60 = n * 360$. $-960 = n * 360$. $n = -960 / 360 = -96/36 = -8/3$. Still not an integer. Let me recalculate my example... Ah, I see the mistake. It's easier to just add or subtract $360^{\circ}$ repeatedly. Let's try that for $-1020^{\circ}$. If we add $360^{\circ}$ multiple times: $-1020^{\circ} + 360^{\circ} = -660^{\circ}$. $-660^{\circ} + 360^{\circ} = -300^{\circ}$. $-300^{\circ} + 360^{\circ} = 60^{\circ}$. We got $60^{\circ}$, not $-60^{\circ}$. So, $-1020^{\circ}$ is not coterminal. Let's try another approach. We want to see if $-1020^{\circ} = -60^{\circ} + n \cdot 360^{\circ}$. Rearranging gives $-1020^{\circ} + 60^{\circ} = n \cdot 360^{\circ}$, so $-960^{\circ} = n \cdot 360^{\circ}$. Dividing by $360^{\circ}$ gives $n = -960/360 = -96/36 = -8/3$. Since $n$ must be an integer, $-1020^{\circ}$ is not coterminal. Okay, let's stick with the $\alpha + 60^{\circ} = n \cdot 360^{\circ}$ method, it feels more solid if I do the math right.

• $-780^{\circ}$: Is $-780^{\circ} + 60^{\circ}$ a multiple of $360^{\circ}$? $-780^{\circ} + 60^{\circ} = -720^{\circ}$. Is $-720^{\circ}$ a multiple of $360^{\circ}$? Yes! $-720^{\circ} / 360^{\circ} = -2$. Since $-2$ is an integer, $-780^{\circ}$ is coterminal with $-60^{\circ}$. This means if you rotate clockwise $780^{\circ}$, you end up in the same spot as rotating clockwise $60^{\circ}$. That's two full turns plus an extra $60^{\circ}$ clockwise!

• $-420^{\circ}$: Is $-420^{\circ} + 60^{\circ}$ a multiple of $360^{\circ}$? $-420^{\circ} + 60^{\circ} = -360^{\circ}$. Is $-360^{\circ}$ a multiple of $360^{\circ}$? Yes! $-360^{\circ} / 360^{\circ} = -1$. Since $-1$ is an integer, $-420^{\circ}$ is coterminal with $-60^{\circ}$. One full clockwise turn from $-60^{\circ}$ gets you to $-420^{\circ}$.

• $60^{\circ}$: Is $60^{\circ} + 60^{\circ}$ a multiple of $360^{\circ}$? $60^{\circ} + 60^{\circ} = 120^{\circ}$. Is $120^{\circ}$ a multiple of $360^{\circ}$? No, $120/360 = 1/3$, which is not an integer. So, $60^{\circ}$ is not coterminal with $-60^{\circ}$. They are different angles, though related!

• $120^{\circ}$: Is $120^{\circ} + 60^{\circ}$ a multiple of $360^{\circ}$? $120^{\circ} + 60^{\circ} = 180^{\circ}$. Is $180^{\circ}$ a multiple of $360^{\circ}$? No, $180/360 = 1/2$, not an integer. So, $120^{\circ}$ is not coterminal.

• $300^{\circ}$: Is $300^{\circ} + 60^{\circ}$ a multiple of $360^{\circ}$? $300^{\circ} + 60^{\circ} = 360^{\circ}$. Is $360^{\circ}$ a multiple of $360^{\circ}$? Yes! $360^{\circ} / 360^{\circ} = 1$. Since $1$ is an integer, $300^{\circ}$ is coterminal with $-60^{\circ}$. This makes sense: $-60^{\circ}$ is the same as $300^{\circ}$ ($ -60 + 360 = 300 $). They point in the same direction!

• $420^{\circ}$: Is $420^{\circ} + 60^{\circ}$ a multiple of $360^{\circ}$? $420^{\circ} + 60^{\circ} = 480^{\circ}$. Is $480^{\circ}$ a multiple of $360^{\circ}$? No, $480/360 = 48/36 = 4/3$, not an integer. So, $420^{\circ}$ is not coterminal.

• $660^{\circ}$: Is $660^{\circ} + 60^{\circ}$ a multiple of $360^{\circ}$? $660^{\circ} + 60^{\circ} = 720^{\circ}$. Is $720^{\circ}$ a multiple of $360^{\circ}$? Yes! $720^{\circ} / 360^{\circ} = 2$. Since $2$ is an integer, $660^{\circ}$ is coterminal with $-60^{\circ}$. This means $660^{\circ}$ is equivalent to $-60^{\circ}$ plus two full rotations ($ -60 + 2*360 = -60 + 720 = 660 $).

The Final Answer: Which Angles Are Coterminal?

After all that checking, guys, we've narrowed it down! The angles that are measures of angles coterminal with $-60^{\circ}$ are the ones where adding $60^{\circ}$ resulted in a whole number multiple of $360^{\circ}$. Let's list them out clearly:

• $-780^{\circ}$: Because $-780^{\circ} + 60^{\circ} = -720^{\circ}$, which is $-2 \times 360^{\circ}$.
• $-420^{\circ}$: Because $-420^{\circ} + 60^{\circ} = -360^{\circ}$, which is $-1 \times 360^{\circ}$.
• $300^{\circ}$: Because $300^{\circ} + 60^{\circ} = 360^{\circ}$, which is $1 \times 360^{\circ}$.
• $660^{\circ}$: Because $660^{\circ} + 60^{\circ} = 720^{\circ}$, which is $2 \times 360^{\circ}$.

So, the checkboxes you should select are:

□ $-1020^{\circ}$ □ $-780^{\circ}$ □ $-420^{\circ}$ □ $60^{\circ}$ □ $120^{\circ}$ □ $300^{\circ}$ □ $420^{\circ}$ □ $660^{\circ}$

Keep practicing these, and you'll be a coterminal angle master in no time! It's all about understanding that full circle rotation. Happy calculating!