Measuring Angles in the Coordinate Plane

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Lesson Priority: High

Pre-Calculus $\longrightarrow$
Trigonometry $\longrightarrow$
  • Look at the standard way in which we measure angles in the coordinate plane
  • Understand why and how negative angles are possible, and interpret how they are different from positive angles
  • Understand why and how angles can be larger than 360 degrees (one full circle rotation)
  • Know what co-terminal angles are and how to tell if two angles are co-terminal
Lesson Description

Having re-remembered all the important facts from the last lesson about measuring angles with degrees, we take our first look at a very common way of working with angles - by putting them in the coordinate plane. Then we can open up angles at any measure we want, and look at the properties that result, such as which quadrant the angle ends up in.

Practice Problems

Practice problems and worksheet coming soon!


Plain Angles to Plane Angles

Most of your experience working with measuring angles has been related to a specific shape, such as examining a triangle that has specified angle measures. Now it's time to bring angle measures to the coordinate plane, which will give us the basis for understanding advanced trigonometry.
I Used To Know That!
Remember that in the coordinate plane, the quadrant on the upper-right side where $x$ and $y$ coordinates are each positive is called Quadrant 1. We then move counter-clockwise to count subsequent quadrants.
Because angle measurement in the coordinate plane has specific uses, we are not allowed to orient an angle measurement at any location we want. If there was no standard in place, then three different students may draw a $30^{\circ}$ angle in three different ways:We need a consistent way to communicate angles, and we call it "standard position".

Standard Position Angles

Every angle we draw is represented as the gap between two lines (or more precisely, two rays). When we measure an angle in the plane, we always keep one of the rays fixed on the positive $x$ axis, and the other one opens in the counter-clockwise direction, so that as an angle gets bigger, it passes through the quadrants in order.
Vocab FYI
When we plot an angle in the coordinate plane, the ray that is drawn along the $x$ axis is called the initial side of the angle, while the other ray is called the terminal side of the angle.
Here is a $45^{\circ}$ angle in standard position:In Geometry it is not unusual to work with obtuse angles for shapes like triangles and polygons. Obtuse angles as we know them in the context of Geometry are angles that are bigger than $90^{\circ}$ and smaller than $180^{\circ}$, such as this $150^{\circ}$ angle:For angles that are even bigger than our classic obtuse angles, we just keep rotating the "free" arm of the angle counter-clockwise:
Pro Tip
I like to think of the mechanics of the angle measurement like a hinge. One arm is "glued" to the $x$ axis while the other is free to rotate.
Example 1Draw $60^{\circ}$, $135^{\circ}$, and $240^{\circ}$ on the same axes in standard position.$\blacktriangleright$ The quadrant boundaries occur every $90^{\circ}$. Using $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ as guides, we can see that a $60^{\circ}$ angle will terminate in the first quadrant, a $135^{\circ}$ will terminate in the second quadrant, and a $240^{\circ}$ will terminate in the third quadrant.
Vocab FYI
The angles that land right on the quadrant boundaries are not said to belong to a quadrant. In fact, we refer to them as quadrantal angles, and we should know them well because they are used as a guide when we count up degrees to plot large angles.

Less Familiar Angle Measures

Now that we're moving toward more advanced trigonometry topics, we'll need to be able to work with angles that are any number of degrees, not just angles that are between $0^{\circ}$ and $360^{\circ}$, including angles that are larger than $360^{\circ}$, and even ones that are negative.If you haven't worked with very large or negative angles before, don't worry - that's typical. There isn't much of a point to studying them before you reach a course on Pre-Calculus and Trig.As we said above, standard position tells us to plot one ray of the angle on the positive $x$ axis, and open up the angle until you have reached the specified measure. This doesn't change even when the angle is larger than $360^{\circ}$ - in that case, we open up an angle "all the way around" and then just keep going.
Example 2Plot $495^{\circ}$ on the Cartesian coordinate plane.$\blacktriangleright$ There are $360^{\circ}$ in a circle, so $495^{\circ}$ means to open up an angle one full circle rotation of $360^{\circ}$, and then go another $495-360$ or $135^{\circ}$.
Describing or drawing large angles is simply a matter of figuring out the remaining degrees after $360$, or after multiple $360$ revolutions if the angle is big enough.
Example 3Plot $1000^{\circ}$ on the Cartesian coordinate plane.$\blacktriangleright$ $1000^{\circ}$ is two $360^{\circ}$ revolutions (totally $720^{\circ}$, and then the remaining $1000-720$ or $280^{\circ}$.

Co-terminal Angles

At this point, it's fair to wonder what the real difference between $1000^{\circ}$ and $280^{\circ}$ is. While they share a common location on the plane, they are not exactly the same angle.Recall that we name the ray on the $x$ axis the initial side and the other ray the terminal side. Two angle measures that share a terminal side have a special relationship in trigonometry, because they do in fact "land in the same place".
Define: Co-terminal AnglesTwo angles are said to be co-terminal if the difference in their angle measures is a multiple of $360$.Visually this means that the terminal sides of the angles will share the same location in the coordinate plane.
For example, $495^{\circ}$ and $135^{\circ}$ from Example 2 are co-terminal because their difference, $360$, is a multiple of $360$. Additionally, $1000^{\circ}$ and $280^{\circ}$ from Example 3 are co-terminal because their difference is $720$ or $360 \cdot 2$.As you'll learn in later lessons, co-terminal angles have the same trig function values » and act entirely like the same angle for all intents and purposes. The sole technicality that makes them different is the number of rotations it takes to travel from $0^{\circ}$ to get there.
You Should Know
Computationally, co-terminal angles will always behave the same. The only time we're really tested on the technical difference between two angles that are co-terminal is during your early trigonometry studies, where quizzes will ask you to draw angles to show your understanding of rotating around the plane the appropriate number of times to be technically correct.

Negative Angles

Because we move counter-clockwise as we plot larger and larger angles, it serves us will to think of rotating the terminal side of an angle clockwise as making the angle smaller. And, as with numbers, when you keep moving in the "smaller direction", you get negative answers once you pass zero.In short, a negative angle is simply one that starts at the initial position but moves clockwise instead of counter-clockwise.
Define: Negative AnglesIn the coordinate plane, an angle in standard position with a negative angle measurement is plotted by beginning measurement at the positive $x$ axis and rotating the terminal side clockwise.
Example 4Plot $-45^{\circ}$ on the Cartesian coordinate plane.$\blacktriangleright$ We will simply move $45^{\circ}$ in the clockwise direction.
You Should Know
Some teachers will begin the unit on advanced angle measurement by introducing a new angle measurement unit called radians, and others will go in the order I'm going in. Either way is fine - the next lesson is about radian angle measure » and we'll practice this co-terminal and negative angle stuff there as well to help us learn the radian measurement system.

Finding Co-terminal Angles

There are situations where we either need or are asked to find co-terminal angles. Fortunately, this is not a difficult thing to remember how to do because the definition of co-terminal is mathematically simple.
Obtaining Co-terminal AnglesSince two co-terminal angles have a difference in their measurements that is $360$ or a multiple of $360$, we can take any angle measurement and add or subtract $360$ to it to obtain a co-terminal angle.
Fun Fact
It shouldn't be surprising that we can create an infinite number of co-terminal angles, since visually, the next co-terminal angle is obtained by rotating any angle around one more full circle, which is an action we could keep doing indefinitely.
Very frequently, we will prefer to work with positive angle measurements. If you're in a special situation that involves a negative angle, we can find a positive co-terminal one by repetitively adding $360$ until the measurement is positive.
Example 5Find a positive co-terminal angle for $-300^{\circ}$.$\blacktriangleright$ We don't even need to graph this to find the answer (though you can if it helps!). We can add $360$ to the angle measurement and obtain a co-terminal angle:$$-300^{\circ} + 360^{\circ} = 60^{\circ}$$Therefore, $60^{\circ}$ is co-terminal with $-300^{\circ}$.

Put It To The Test

See if you can replicate what you just learned with some quick examples below!
Example 6Plot $240^{\circ}$ in standard position in the coordinate plane.
Show solution
$\blacktriangleright$ Use your quadrant dividing angles to determine that $240^{\circ}$ is bigger than $180^{\circ}$ and smaller than $270^{\circ}$. Specifically, it is $30^{\circ}$ shy of reaching $270^{\circ}$.
Example 7Plot $-100^{\circ}$ in standard position in the coordinate plane.
Show solution
$\blacktriangleright$ The only difference between plotting a positive angle or a negative one is the direction of rotation. For this angle we must travel $100^{\circ}$ from the initial side in the clockwise direction.
Example 8Plot $1200^{\circ}$ on the coordinate plane.
Show solution
$\blacktriangleright$ When it comes to extraordinarily large angles, the idea is to figure out two things: first where we are landing (the reasonably sized co-terminal angle), and second, how many rotations we need to get there.The angle co-terminal to $1200^{\circ}$ is found by subtracting $360^{\circ}$ a few times.$$1200^{\circ} - 360^{\circ} = 840^{\circ}$$$$840^{\circ} - 360^{\circ} = 480^{\circ}$$$$480^{\circ} - 360^{\circ} = 120^{\circ}$$Therefore we have figured out not only that $1200^{\circ}$ is co-terminal with $120^{\circ}$, but that it is three full rotations past $120^{\circ}$ (since we subtracted $360^{\circ}$ three times to get to $120^{\circ}$).
Example 9Find two co-terminal angles to $45^{\circ}$ in the coordinate plane - one positive and one negative.
Show solution
$\blacktriangleright$ Recall that co-terminal angles are obtained by adding or subtracting $360^{\circ}$ from the angle you're looking at. In this case, we'll add $360^{\circ}$ to obtain a positive co-terminal angle to $45^{\circ}$ and subtract $360^{\circ}$ to obtain a negative co-terminal angle.$$45^{\circ} + 360^{\circ} = 405^{\circ}$$$$45^{\circ} - 360^{\circ} = -315^{\circ}$$If we want or need to plot these co-terminal angles, remember that they will "land" in the same spot as $45^{\circ}$ but demonstrate the direction and exact amount of rotation.
Lesson Takeaways
  • Understand how standard position angles are oriented in the coordinate plane
  • Be able to plot positive, negative, and large angles in the plane
  • Know how to find co-terminal angles as requested

Lesson Metrics

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