# Reference angle practice

In the figure above, as you drag the orange point around the origin, you can see the blue reference angle being drawn. It is the angle between the terminal side and the x axis. As the point moves into each quadrantnote how the reference angle is always the smallest angle between the terminal side and the x axis.

Regardless of which quadrant we are in, the reference angle is always made positive. Drag the point clockwise to make negative angles, and note how the reference angle remains positive. Drag the point around the origin several times. For negative angles add instead. Sketch the angle to see which quadrant it is in. In trigonometry we use the functions of angles like sincos and tan. It turns out that angles that have the same reference angles always have the same trig function values the sign may vary.

The rest we can find by first finding the reference angle. Home Contact About Subject Index. Definition: The smallest angle that the terminal side of a given angle makes with the x-axis. Try this: Adjust the angle below by dragging the orange point around the originand note the blue reference angle.In the previous section, we found the first-circle angle equivalents for given angle measures. Another thing we can do with angle measures, even those whose measures are in the first go-around, is to find what is called the "reference" angle.

The reference angle is the angle that the given angle makes with the x -axis. Regardless of where the angle ends that is, regardless of the location of the terminal side of the anglethe reference angle measures the closest distance of that terminal side to the x -axis. Let's get started with an easy example. For graphing, the angle's initial side is the positive x -axis; its terminal side is the green line, because angles are drawn going anti-clockwise. The curved green line shows the given angle.

The reference angle, shown by the curved purple line, is the same as the given angle. Notice how this last calculation was done.

I didn't have a graph. I just did the arithmetic in my head. You should draw graphs for as long as you need the help, but don't be afraid to start relying on the arithmetic. Once you get the hang of this, it's really pretty straightforward. Note: Because the reference angle always measures the positive distance from the x -axis, it can also be viewed as being the first-quadrant equivalent angle.

In other words, for each of the examples above, if my textbook defined "reference angle" as "the first-quadrant angle with the same distance from the x -axis", then the purple "reference angle" line the curved purple line, plus a terminal side would have been drawn in the first quadrant.

Either way, the value for the reference angle will always be the same. But if you are required to draw a picture showing the reference angle, make sure you draw it in the location that's regarded as "correct" for your class. That means that the left-over portion the 0. This is smaller than ninety degrees, so the terminal side of the angle is to the right of the positive y -axis. Then the reference angle is in the first quadrant and is equal to:.

Two cycles fit within the angle. How much of the angle's measure do those two cycles take up? This angle is between those values, so it's in the third quadrant, and will be closest to the negative x -axis.

## Reference angle - Trigonometric Ratios and Angle Measures

How close? It'll be the distance between the terminal side of the reduced angle and the negative x -axis:. Notice how I drew the reduced angle being the original angle, less two cycles in green, and then I drew the first-quadrant reference angle in purple. When you're doing drawings that contain two or more distinct pieces of information, it can be helpful to have colored pencils on hand.

Yes, I used colored pencils in college. Okay, this is in radians. In other words, this angle goes a little past the negative x -axis:. But how far is the terminal side from the negative x -axis? I can figure this out by subtracting the angle measure of the negative x -axis from my reduced angle:.Our reference angle calculator is a handy tool for recalculating angles into their acute version. All you have to do is simply input any positive angle into the field and this calculator will find the reference angle for you.

This article explains what a reference angle is, providing a reference angle definition. It will also provide you with a step-by-step guide on how to find a reference angle in radians and degrees, along with a few examples. Keep scrolling, and you'll find a graph with quadrants as well! Look at the picture above. Every angle is measured from the positive part of the x-axis to its terminal line the line that determines the end of the angle traveling counterclockwise.

If you want to find the reference angle, you have to find the smallest possible angle formed by the x-axis and the terminal line, going either clockwise or counterclockwise. Reference angles are useful in trigonometry. If you want to find the sine or cosine of any arbitrary angle, you first have to look for its reference angle in the first quarter. Then you can find the trigonometric function of the reference angle and choose a proper sign. The two axes of a 2D Cartesian system divide the plane into four infinite regions that are called quadrants.

Numbering starts from the upper right quadrant, where both coordinates are positive, and goes in an anti-clockwise direction, as in the picture. Generally, trigonometric functions sine, cosine, tangent, cotangent give the same value for both an angle and its reference angle. The only thing that changes is the sign - these functions are positive and negative in various quadrants.

Make sure to take a look at our law of cosines calculator and our law of sines calculator for more information about trigonometry. Reference Angle Calculator can be embedded on your website to enrich the content you wrote and make it easier for your visitors to understand your message.

Get the HTML code. Omni Calculator logo Embed Share via. Reference angle. Check out 53 similar 2d geometry calculators. Table of contents: What is a reference angle? Reference angle definition Graph quadrants and trigonometric functions How to find the reference angle for degrees How to find reference angle in radians How to use this reference angle calculator What is the reference angle for What is a reference angle? Reference angle definition Look at the picture above.

Graph quadrants and trigonometric functions The two axes of a 2D Cartesian system divide the plane into four infinite regions that are called quadrants.

A for all : in the first quadrant, all trigonometric functions have positive values. S for sine : in the second quadrant, only the sine function has positive values. T for tangent : in the third quadrant, tangent and cotangent have positive values. C for cosine : in the fourth quadrant, only the cosine function has positive values. Keep doing it until you get an angle smaller than a full angle.

This is the same as finding the modulo. How to find reference angle in radians It's easier than it looks! How to use this reference angle calculator It couldn't be easier! Simply: Type the angle into the box. If you want, you may also change the units, e. Thats it!The reference angle is always positive. In other words, the reference angle is an angle being sandwiched by the terminal side and the x-axis.

It must be less than 90 degree, and always positive. Knowing these, you can use them as shortcuts to help you determine around where a reference angle should be and if you're in the right range. Reference Angle: the acute angle between the terminal arm and the x-axis; reference angle is always positive. We want to determine the reference angle of a degree angle. First we draw the standard angle of degrees on a xy plane.

Starting from the x axis zerowe turn the terminal arm to the positive direction since we are dealing with a positive angle. We stop at degree and get the standard angle. From the graph, we know that the angle lands on the second quadrant or quadrant 2.

Now, we can determine the reference angle. Based on the definition of a reference angle, we can determine that the reference angle is 50 degree. In this question, we are looking for the reference angle of degree. Same as the last example, we draw the standard angle of degrees on a xy plane. Starting from the x axis zerowe turn the terminal arm to the positive direction.

We stop turning the terminal arm when it reaches degree, and we get the standard angle. Now, according to the definition of a reference angle, we can determine that the reference angle of degree is 20 degree.

And we know that the standard and reference angle land on the third quadrant or quadrant 3. This time, we are going to find the reference angle of a negative angle: degree.

Same as the last example, we draw the standard angle of degree on a xy plane. Also, starting from the x axis zerohowever, this time we turn the terminal arm to the negative direction. Now, according to the definition of reference angle, we can determine that the reference angle of degree is 23 degree.

As a refresher, it's just another way to measure angles other than degrees. Do also revise coterminal angleswhich are also angles in standard positions with a common terminal side. Back to Course Index. You can still navigate around the site and check out our free content, but some functionality, such as sign up, will not work. If you do have javascript enabled there may have been a loading error; try refreshing your browser. Still Confused?The reference angle is the positive acute angle that can represent an angle of any measure.

Basically, any angle on the x-y plane has a reference angle, which is always between 0 and 90 degrees.

Unit Circle Definition of Trig Functions

The reference angle is always the smallest angle that you can make from the terminal side of an angle ie where the angle ends with the x-axis. A reference angle always uses the x-axis as its frame of reference. Positive angles go in a counter clockwise direction. Below is a picture of a positive fifty degree angle. Every positive angle in quadrant I is already acute Remember that the reference angle always uses the x-axis as a frame of reference.

Trigonometry Gifs. What is the reference angle for the angle in the graph below? Show Answer. Further Reading: Unit Circle Game free online game on all things about the unit circle Unit Circle Printables images of blank unit circles and blank unit circles with the answers filled in Unit Circle Worksheet.

Popular pages mathwarehouse. Surface area of a Cylinder. Unit Circle Game. Pascal's Triangle demonstration. Create, save share charts. Interactive simulation the most controversial math riddle ever! Calculus Gifs. How to make an ellipse. Volume of a cone. Best Math Jokes.Show Step-by-step Solutions Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

### Reference Angle Calculator

Related Topics: More Lessons for Trigonometry Math Worksheets In this lesson, we will look into how to use the reference angle to find the sine, cosine and tangent of non-acute angles.

What is a reference angle? A reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. How to find the reference angle? Step 1: Sketch the given angle Step 2: Drop a perpendicular to the x-axis Step 3: Determine the angle measure of the triangle formed How to use reference angles to find the sine, cosine and tangent of non-acute angles?

To find the value of sine, cosine and tangent at non-acute angles from 90 tofirst draw the angle on the unit circle and find the reference angle. A reference angle is formed by the terminal side and the x-axis and will therefore always be acute. When evaluating sine, cosine and tangent for the reference angle, determine if each value is positive or negative by identifying the quadrant the terminal side is in. Show Step-by-step Solutions. You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics.

Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.And besides, this could end up being a borderline home game for the Eagles. The network broadcasting the game is hiring Rams fans to show up for pre-game, for crying out loud. It wasn't easy to pick the Eagles in this one, especially since I picked them a week ago and was wrong. But before the team embarked on this current two-game swing out west, I was fairly confident that they would return to Philly on Monday with a 1-1 record in those two games. It just turns out that I picked the wrong game.

Had the Birds won in Seattle, their inevitable reality check would've come this week in L. But since that wasn't the case, I think its safe to say the Eagles will bounce back this week at the Coliseum, which will likely be filled with more fans wearing midnight green than blue and gold.

I don't believe the Rams defense will be good enough to keep the Birds high-flying offense in check, so the real matchup in this one will likely be between the Eagles defense and the Rams offense. And while Goff and Wentz will be getting most of the headlines, the real difference-maker on Sunday could be Todd Gurley, who is currently on pace for over 2,000 yards from scrimmage this season.

One way the Eagles offense can help limit Gurley's damage is by getting an early lead and forcing Goff to throw on them, something they were unable to do against Russell Wilson and the Seahawks. MORE ON THE EAGLESWeek 14 NFL predictions: Rounding up the experts' picks for Eagles-RamsFOX seeks actors to be Los Angeles Rams fans for Eagles gameMailbag: Where does Doug Pederson rank among the 2016 NFL coaching hires.

Zach Ertz returns to practiceNot only will the game feature a matchup of two of the top teams in the NFC, it will also be the first head-to-head matchup between Rams quarterback (and former No. In other words, expect a tight game on Sunday, one in which almost anything can happen. See this section for more details. Make sure to use the full path if the file is not in your current directory.

As with all BigML. You can get jq here. In the Datasets Section you can learn how customize the parsing rules and other options when converting a datasource to a dataset. Each field in your source is automatically assigned an id that you can later use as a parameter in models and predictions. In the Models Section you will learn how to customize the input fields or the objective field. You can use this id to retrieve the prediction later on. The predicted value is found in the prediction object, keyed by the corresponding objective field id.

A quick start guide for the impatient is here. You can use BigML. That is to say, using BigML. Fully white-box access to your datasets, models, clusters and anomaly detectors.

Asynchronous creation of resources. The four original BigML resources are: source, dataset, model, and prediction. As shown in the picture below, the most basic flow consists of using some local (or remote) training data to create a source, then using the source to create a dataset, later using the dataset to create a model, and, finally, using the model and new input data to create a prediction.

The training data is usually in tabular format. Each row in the data represents an instance (or example) and each column a field (or attribute). These fields are also known as predictors or covariates. When the machine learning task to learn from training data is supervised one of the columns (usually the last column) represents a special attribute known as objective field (or target) that assigns a label (or class) to each instance.

The training data in this format is named labeled and the machine learning task to learn from is named supervised learning. Once a source is created, it can be used to create multiple datasets. Likewise, a dataset can be used to create multiple models and a model can be used to create multiple predictions. 