17 Free Homework Help Websites for Math, Coding, Chemistry, Music, Languages, and English

homework help websites

Are you drowning in homework?

Finding a reliable and trustworthy homework help website is not always easy.

But by the time you finish reading this article, my goal is for you to have some valid homework help options in place.

I’ve put together a list of homework help websites that are free to access and easy to use.

There’s just one thing.

None of them involve interacting with people.

Now don’t get me wrong!

Passively interacting with a website can be helpful, and these homework help websites are LEGIT.

I’m taking a pretty liberal approach to this term homework help.

Contract cheating companies offer homework help and so does the Khan Academy, but these are vastly different approaches to your homework.

Of course, any type of homework help will succeed to a limited extent.

However, sometimes it is easier to start a chat with someone when you really want to learn something.

Or maybe you just want answers. In which case it would be fastest to click the button below.

Get Answers Now

The content you will find below is organized in two ways.

First, you can also view the resources organized by a little secret of Internet marketers called a DA (domain authority) score.

Every website gets a DA score from 0-100. Twitter is 100. A freshly minted blog will be somewhere between 0-5.

Think of it like a digital popularity contest where bots get to decide who wins.

The number depends on a website’s ability to offer valuable targeted content to its users.

This will help you judge the TRUE value of the website you’re considering and not just the hype.

homework help websites

Second, I take into account the kind of homework help resource offered. For example, is this homework help website intended for learning, giving answers, or offering resources?

This question really matters since all of the websites below give homework help.

Let’s get off to the organized list.

Homework Help Websites for Learning

YouTube | DA: 100

YouTube gets tremendous gamer and entertainment traffic, but it turns out that YouTube has some practical learning applications too.

Because of the amazing and entertaining education channels. Chemistry with Melissa Maribel, Patrick JMT, and Physics Girl are but a few of the world class online educators.

The keyword emphasis there is “entertaining.” Their job is to entertain you first, and if they educate you too that’s great.

If you want to learn on YT, it will be way more efficient to search for specific terms than browse by channel.

Wikihow | DA: 92

Wikihow is in-depth and step-by-step, but don’t expect all of the questions to be educational.

Sure, you can get help solving quadratic equations. And you can also get help with how to kiss in a car.

Either way, Wikihow will give you graphically designed no-BS straight to the point answers.

Green check marks on the article mean the answer has been reviewed by an expert with a doctorate, and the most of the content gets edited by at least 23 people.

Khan Academy | DA: 91

Everyone knows about the Khan Academy.

What some people don’t know is that they have an app for 2-5 year olds.

If you’re a gamer, KA works great as it measures energy points and lets you earn badges.

homework help websites

Besides their fame for math and science, KA also has art videos of professors discussing famous paintings at museums.

KA enjoys a second to none reputation in expanding a world education to everyone for free.

But you won’t get answers from them.

Codecademy | DA: 87

If you want to learn coding for free, you can use the entire Codecademy site for 7 days.

After that, the site will force you to convert into a paid user.

You’ve been warned!

Languages to learn include HTML & CSS, Python, JavaScript, Java, SQL, Bash/Shell, Ruby, and C++. You’ll find world-class content here.

Sparknotes | DA: 81

If you haven’t heard of Sparknotes, you just discovered the most powerfully useful English Literature resource on the Internet.

They serve way too many ads, but they know their stuff and deliver detailed content about popular literature works.

Their fantastic No Fear Shakespeare resource places the original 16th-17th century text alongside modern translations.

That makes it a whole lot easier to understand.

Duolingo | DA: 81

Duolingo offers free AI-based learning for complimenting your language studies in class.

Don’t expect it to singlehandedly supplement your classwork in the given language since it can’t be customized to fit in specific vocab words.

But you CAN focus specific language areas like phrases, plurals, and vocabulary areas (food, home, etc.).

I haven’t found a way to focus on specific parts of grammar.

Instead, it will measure your competency and then put you through its program based on how you test.

Ableton Learning Music | DA: 69

The beauty of Ableton comes its extremely interactive interface.

You can click everything. This website is a free candy store for making sound.

Best of all, you immediately hear the music. This way there is no chance that the concept will get lost in abstract terminology.

homework help websites

It’s also easy to click through the site without understanding what is happening.

Be prepared to play with this site to help master something specific like chord inversions and you will do well.

Anki | DA: 55

Anki works as a series of flashcards to help you memorize the topics that you are trying to learn.

You can make your own flashcards.

There are also free flashcards, and this option works best for foreign language and science (especially pre-med) students.

The app costs $25, but you can use it for free on your laptop.

Honestly, I bought the app and I don’t think it is worth it since I never study on my phone.

Homework Help Website Answers

Reddit | DA: 98

The r/HomeworkHelp subreddit serves lots of answers.

The mods specifically ban “do this for me” posts, but I still see many direct questions.

This is definitely worth a shot if you have something basic and straightforward to offer.

Quora | DA: 92

The following pages have strong followings on Quora and are listed in order of popularity:

  1. Mathematics Homework Questions
  2. Homework Questions
  3. Chemistry Homework Questions

Quorans tend to be a little snarky about giving away answers, but there are people there who want to help.

Even better than Reddit, you can check their account to see their background.

Like YouTube, I would recommend to search by your question.

The search algorithm is tougher to use than YT, but with some key word alterations you should be able to find what you’re looking for.

Wolfram Alpha | DA: 86

Stuck on an equation? Wolfram Alpha covers math as well as science and technology, society and culture, and everyday life.

Think Wikipedia for Math and you get how this site works.

The designers of Wolfram Alpha do a terrific job. The site offers facts, numbers, and dates in as few words as possible.

However, this is a paid service if you are looking for work shown.

Homework Help Website Resources

Google Scholar | DA: 91

Google scholar helps you find the articles that you want later on in their full form (see Sci-hub).

I highly recommend showing results based on date of publication and sorting by relevance.

Many teachers don’t like sources that are older than five years.

Once you find what you want, make sure to copy+paste that DOI somewhere.

Grammarly | DA: 77

Grammarly will edit your paper for the following:

  • Spelling
  • Grammar
  • Punctuation
  • Conciseness
  • Formality
  • Sentence Variety

It will even tell you if you plagiarized portions of the paper.

homework help websites

However, the plagiarism checker is not as accurate as Turnitin since it does not check other papers and journal articles.

The editing, however, works at a high quality level.

Sci-Hub | DA: 55

Sci-hub is a Russian website that lets you download academic papers for free.

It’s also controversial because it mass distributes paid content.

So will you get in trouble for using this? In the USA, the copyright owner can sue (typically for money damages or injunctive relief) an infringer.

The criminal laws are aimed at the reproducer and/or distributor, however.

Also, according to Pablo Markin at Open Science, publishing company Elsevier got blocked in Sweden after it legally required internet service providers to make Sci-Hub locally inaccessible.

Why use Sci-hub in the first place? Speed and fairness.

If you have the DOI of an article, you can usually find it 10x faster on Sci-hub than you can on any online database.

If you consider that professors do not make any money off of the articles they publish anyway, using Sci-hub makes a lot of sense.

LibGen | DA: 55

Much of the above commentary on Sci-hub applies here.

Both websites originate in Russia.

Both offer users copyrighted content and mass distribute it.

In this case, the content consists of books, not articles. You’ll want the ISBN number ideally, but you can always use the title to find an alternate edition.

Besides saving money on your textbooks, Libgen contains access to millions of books you can get in PDF and EPUB formats.

Symbolab | DA: 49

This is a free graphing calculator if you don’t have yours or don’t own one.

It is also offered as an app for Android and iOS, which provides an additional layer of convenience.

They even give you a Cheat Sheet of formulas in Algebra, Trigonometry, Limits, Derivatives, and Integrals.

You can download it in PDF form: get the cheat sheet 🙌

MMSPhysChem Stoichiometry Calculator | DA: 28

This is the simplest site on the list, but that might make it one of the most directly useful too.

Look no further if you want help on stoichiometry calculations.

Redox reactions, limiting reactant problems, and chemical equations containing hydrates can not be solved using this system at this time.

So that’s it for my free options.

If you’re still wanting to truly deepen the grasp of your homework, Studygate can help with that.

homework help websites
The tools you need to succeed, right at your fingertips
  • Access 1,000 on-demand tutors. We sourced them to make sure they are experts in their fields.
  • Get 24/7 help since we’re here to provide homework help anytime, day or night.
  • Enjoy lightning-fast responses for homework help within minutes.
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You can get a tutor to chat about your homework and help answer your question directly. It’s that simple.

Need Homework Help? Click Here!

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How to Measure a Central Angle

Concept of Central Angle

Three Little Kittens

Found their Mittens

So they shall have some pie.

Said one, “The size for me

is an arc from A to B

andjust shy of 360!”


Not quite sure what this little kitten was going on about?

Well, the central angle is one way we can use to measure circles.

As the name suggests, it is an angle with the vertex at circle’s center. Like this:

central angle

Here, we can see that the vertex is the center of the circle, with points A and B on the outer edge. Each side of angle ∠AOB is a radius, which means that it is half of the diameter.

The outer edge of the circle between points A and B—the kitten’s pie crust—is an arc. The arc length is the measurement between A and B along the curve.

Okay, now that we know the terms cold, let’s set up some relationships.

We can make up some ratios. The arc length from AB is a portion of the entire circumference. There is a direct ratio between the circumference of a circle and its measurement of degrees. The A circle has 360°. The central arc is a portion of that.

central arc

Let’s say we know the arc length was 25π and the circumference was 166π. We can calculate the central angle easily!

calculating central angle

What if we only knew the length of one side of an angle and the arc length?

This is a bit more challenging, but it’s just an extra step.

Let’s recap with our original angle & circle:

central angle

Say OA was 6 feet. Arc length is 1.4π

We can calculate the circumference of the circle: Circumference = Diameter x π

The radius is half the diameter, and we know that AO is a radius because it extends from the center of the circle at point O and reaches the outer edge at point A.

Circumference = Diameter x π = 2(radius) x π
Circumference = 2(6 feet) x π
Circumference = 12π

Great job! Now, we can plug in the numbers into our original formula!

central angle formula

Yep, pretty small angle, all right. Let’s hope that’s not the one our kitty ended up with! Stay tuned to keep learning on online tutoring platform StudyGate.

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When, Why, and How to Use the Half-Angle Formula

The half-angle formula, and its counterpart, the double-angle formula, will usually crop up in trigonometry, and sometimes pre-calculus classes. Sometimes you’ll be instructed to evaluate a trig function of an angle that’s not specifically included on the unit circle, and by using the identity of an angle that is included, you can still evaluate the function. For example, if you’re asked to find sin105º, an angle that’s not on the unit circle, you can use a half-angle identity and substitute 210º, which is on the unit circle, for 105º. The half-angle formulas for sine, cosine, and tangent are as written:

half angle formula

Follow this step-by-step process to use the half angle formula successfully.

If we’re trying to find sin105º, we first have to recognize that 105º is half of 210º, which is featured on the unit circle. We’d then rewrite the function as sin(210/2). Notice that the sine equation has the ± symbol, so we need to determine whether this function will be positive or negative. 105º lies in the second quadrant of the unit circle, so it will be positive. We then substitute x for the full angle value of 210. We then look to the unit circle again to find the cosine value of 210º, which is -3/2, and substitute that value for cos210. The final step is to simplify the function if possible. In this case, we can, because the denominators can be simplified.

While the formula seems tricky at first, it can be incredibly helpful in evaluating angle functions that are not listed on the unit circle. Always remember to be mindful of your plus and minus signs, as a function can transform into something completely bizarre and unintended otherwise. It also pays to check your simple addition, subtraction, and multiplication; mastering the formula isn’t worth much if you mix up your simple equation procedures. With the right amount of practice, the possibilities are limitless! StudyGate provides online learning resources to attain academic excellence.

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Angle Addition Postulate

angle addition postulate example

Okay, here’s a brain zinger: if someone tells you that they are using the awesome Angle Addition Postulate, what do you think they are adding?

If you said “angles,” you would be 100 percent correct!

The Angle Addition Postulate basically means we are taking two angles and joining them together to make one LARGER angle!

Here’s a basic example”

We’ll take ∠GEM and ∠MEO

  • Let’s join them together do that the angle converges at point E.
  • We’ll also lay them together so that the two angles share the same border EM.

angle addition postulate definition

What can we do with the Angle Addition Postulate?

Great question!

Let’s look at the above angle again. If ∠GEO was 125°, then what is ∠GEM?

It’s almost like slicing a piece of cake!

∠GEM + ∠MEO = ∠GEO

∠GEM + ∠MEO = 125°

We know that ∠MEO = 90° because it is a right angle with the little square.

∠GEM+ ∠90° = 125°

∠GEM = 35°

Let’s go for something harder:

What are the angles below?

angle addition postulate example


First of all, we know that ∠ABD and ∠DBC follow the Angle Addition Postulate. This is because they share the same vertex at point B. The two angles share the same line BD, so they line up.

∠ABD + ∠DBC = ∠ABC

We know that AC is a straight line, and straight angles are 180°

We also have the formulas for ∠ABD and ∠DBC

Let’s plug ‘em in:

∠ABD + ∠DBC = ∠ABC

(5x+10) + (2x-5) = 180°

5x + 10 + 2x – 5 = 180

7x + 5 = 180

7x = 175

x = 25

So, now that we know what  is, we can plug it into the original formulas! ∠ABD = 5x + 10, x = 25


∠DBC = 2x – 5, x = 25


Let’s check:

∠ABD + ∠DBC = ∠ABC

We are balanced!

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Lord of the 45-45-90 Triangle

45-45-90 triangles: geometry math online learning

Triangles come in all sizes. Some are acute, some are obtuse, and some are just right.

The ones that are just right weren’t designed by Goldilocks. But she knew a right triangle when she saw it. She knew that a triangle was “right” when it had a 90° angle—perfect for drawing sharp corners. Like these:

Right Triangles

But there was an even more special right triangles. She could hardly wait to contain herself when she saw it. It was the awesome 45 45 90 triangle. Here it is in its mighty glory:

45-45-90 triangles: geometry math online learning

What’s so spectacular about this triangle? So many things it’s hard to list them all!

  • The two sides making up the 90° angle have the same length
  • The triangle is half of a square, as you can see on the right.
  • That means its hypotenuse (the diagonal of the square) is the side x √2
  • trigonometric functions



Pretty easy, huh?!

  • The tangent of 45-45-90 triangles is always 1.

tangent of 45-45-90 triangle

Can’t get any easier than that!

Let’s recap!

45-45-90 triangle: geometry math online learning

The angles are easy to remember: 45+45+90 = 180°

The sides and hypotenuse are easy: the two sides are the same! x = x

The Hypotenuse is x√2

sin cos tan in 45-45-90 triangle

Goldilocks is a Lord of the 45-45-90 Triangles.

And now, you are, too!!!

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Angling for Angle Pairs

corresponding angles

There’s something to be said about angles:

“To think I will never see

An angle as complementary as thee.

An angle whose hungry mouth is open

To a supplementary pair unbroken

With a line, perpendicular or straight,

We a newborn angle doth create!”

While poets may sing the praise of angles, let’s set it down in basic terms.

  • What’s an angle?

An angle is the measurement of degrees between two lines.

These are all angles. They are acute angles because the gaps between the two lines are less than 90o. Acute means small, and small things are cute. Aren’t they cute?

acute angles

These are angles, too. They are obtuse angles because the gaps between the two lines are GREATER than 90o. Obtuse means “slow” and “unintelligent.” While we don’t want to be mean to our angles, we can think of them as a bit unwieldly and hard to manage. Why are these angles so obtuse, darn it?!?

obtuse angles

This means that angles that are EXACTLY 90o are called right angles. As Goldilocks would tell us, not too large, not too small, just right. We can make little squares inside them to show that the corner is 90o sharp. Pretty cool, huh?

right angles

OKAY, now that we have that downpat, what do we do with our happy angle family?

  • We can use angles to measure degrees.

For designers like architects, builders, and engineers, this is essential stuff…and for obvious reasons. If you’re measuring space and distance, whether for landscaping or for algorithms, angles are a must.

Let’s throw some basic calculations on how to measure angles.

  • Adjacent Angles

These are pretty simple. It’s literally two angles next to each other! They share a common vertex (the point where they intercept) and one side. Here’s one:

adjacent angles

For example, we see that the blue angle,∠BAD, has a measurement of 58o. The smaller angle ∠BAC has a measurement of 32o.  What is the measurement of ∠CAD?

Simple subtract!

58o – 32o = 26o

Easy, right?

  • Complementary Angles

A complementary angle is a specific type of adjacent angle. These angles add up to 90o to form a RIGHT ANGLE.  Here’s one:

complementary angles

Since we know that the sum of the angles is always 90o, we can easily find out what one angle measurement is if we know the other. Since both angles are smaller than 90o, we know that all angles in complementary angles are acute.

  • Supplementary Angles

Supplementary Angles take things a bit further. When we put two COMPLEMENTARY ANGLES together, they form a straight line, like this:

supplementary angles

See how CO bisects DA? The angles on the right of ∠COA add up to 90o, right? That means that ∠DOC is also 90o. Put them together and we have supplementary angles.  Angles that are supplementary share the same vertex and one line. They form a straight line and add up to 180o. Dead on arrival, no?

Here’s another example of a supplementary angle, this time, without complementary angles.

supplementary angles 2

Note that if one angle is less than 90o, the other must be greater than 90o to add up to 180o. Thus, unless the angles are right angles, supplementary angles include one ACUTE and one OBTUSE angle.

We can also complicate this by stating that in a circle, all the angles add up to 360. This is because two right angles sharing a vertex forms a straight line, but travels half a circle. If we join two more right angles at the same vertex, the degrees of measurement would go all the way around! Like this:

way around angles

Got it? Right on target!

Now let’s mess it up a bit with multiple sets of angles sharing one line, kinda like a road map. Line Y is the transversal line because it joins Lines A and B by crossing them:

supplementary angles 3

We can see that [D] and [E] are supplementary since they share the same vertex, one line (Line Y) and add up to 180o. What else can we say?

  • Corresponding Angles

[E] and [Q] are corresponding angles because they share the same exact position on their parallel lines A and B. Their angles are the same. The same with [F] and [R], [D] and [P], and [G] and [S]. Let’s highlight them:

corresponding angles

See how the angle pairs correspond. What else can we say?

  • Alternate Interior Angles

We also have Alternate Interior Angles. Being interior angles, they are on the “inside,” between the parallel lines. These angles share the central transversal line, but have two different vertices. Think of them forming a letter Z and a reverse Z. In the example above, [G] and [Q] are alternate angles. So are [F] and [P]. Since Lines A and B are parallel,we can see that alternate angles have the same degrees.

alternate interior angles

  • Vertically Opposite Angles

From Z to X!They are angles that share the same vertex and transversal, but no corresponding side. They also have the same degrees. Think of them as the Letter X, with the angles opposite each other. In the above example, [E] and [G] are Vertically Opposite Angles. They each have 80o. So are [D] and [F]. Each one has 100o. Check ‘em out below:

vertically opposite angles

Now that you’re seeing crosses, we’ll have a quick quiz. Find all the angles below:

calculate all the angles

We only have a single degree measurement, 111o, but this is no problem!


A pair of supplementary angles add up to 180o, right? We also know that 111o is OBTUSE, so its adjacent angle, ∠C, must be ACUTE!

∠C + 111o = 180o

∠C = 69o

From there, we can fill out the top set of angles pretty easy.

∠A is a Vertically Opposite Angle from 111o. We know vertically opposite angles have the same degree measurement. So ∠A is 111o

The same with ∠C and ∠B.

∠C is 69o so ∠B is also 69o.

To check, we can also ADD ∠A + ∠B to see if they add up to 180o. (They do!)

We can simply copy and paste the top set of angles of the bottom, but we also use our vast knowledge of angle pairings to figure ‘em out. You’re an expert—go right ahead!

∠G is a corresponding angle with 111o.

That means ∠G =111o

∠D is an alternate interior angle with 111o. They form the inverse Z shape!

That means ∠D =111o

We can either figure out what ∠E and ∠F are since we know their supplementary angles, which add up to 180o

Or, we can look at them in pairs.

∠E is a corresponding angle with∠B. Since ∠B = 69o, ∠E is also 69o.

∠F is a Vertically Opposite Angle from∠E. Since we just figured out what ∠E is, and we know vertically opposite angles have the same value, ∠F is also 69o.


∠A = 111o.

∠B = 69o.

∠C = 69o.

∠D = 111o.

∠E = 69o.

∠F = 69o.

∠G = 111o.

Congrats! You are now the Awesome Angle Master! Have a glimpse of Reference Angle.

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Trapezoid Training

calculating area of a trapezoid

If your first thought of a trapezoid was either a flying circus act or a muscle in the shoulder, this blog is for you!

First, a minor let down. A trapezoid is nothing exotic as double summersault through the air. But it is pretty nifty. This is a trapezoid:
What’s a trapezoid, then? It’s a shape with four sides. Of the four sides, two of them are parallel.

In this case, the top and bottom lines are parallel. The side lines will eventually intersect if they continue downward.

A trapezoid’s side lengths don’t matter. Two sides can be the same, as shown above. Or they can all be different, like these:

trapezoid training online

The top and bottom lines are still parallel, even though no two lines have the same length.

***Burning Question: Is a rectangle a trapezoid? Nope. A trapezoid can only have ONE set of parallel lines. Rectangles (and squares) have two sets! (Now ask yourself if a square can be a rhombus. To find out: see my next blog! 😊)

***Fun note: There is also a shape called a TRAPEZIUM, which is a four-sided shape with no parallel sides, like this:


This is not a trapezoid.

Now that we have a trapezoid, what do we do with it? How can find its area?

Why do we need to find its area? It’s like the old saying goes:

There was an old woman who lived in a shoe

She had so many sons, she didn’t know what to do.

So she sliced up the shoe, like a loaf, and gave them to her boys

“This is your home, enjoy the trapezoid!”

So, how much living space did each kid have?

Let’s say this is the cross section of the shoe slice:

area of a trapezoid online learning

How to figure out the area?

An area for a trapezoid is the SUM of the two parallel sides, divided by 2, and then MULTIPLIED by the height. The height is a perpendicular line between the two parallel sides.

Here we go:

trapezium area formula

So let’s give some dimensions to the shoe:

calculating area of a trapezoid

Let’s plug in those values:

finding trapezoid area

Don’t forget this is squared meters since we are considering two dimensions! We multiplied the labels to get meters2

Fun addition: volume!

But the shoe slice also has three dimensions: length!

It is very easy to figure out the volume of a trapezoid! Simply multiply the area by its length!

trapezoid volume formula

Since we already know the area, let’s just say that the generous old woman gave each son a slice of shoe that was two meters long. Plug in the numbers again:

calculating trapezoid volume

*Note that when measuring by volume, we are dealing with three dimensions, so the figure is cubed!

Hip hooray!

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Coterminal Angles Quickly Solved

central angles

All right, math wizards, here’s a trick question: does a circle have more than 360° in it?
The answer: nope. Once you go all the way around a circle, you end up back at the beginning! Pretty easy, right?
But that doesn’t mean an angle can’t have more than 360°. Here’s an example:

360 degree angle

Say we’re on a Ferris Wheel. If the ride’s any good at all, you just don’t make one trip around and then get off. You keep on going for several rounds, right? For each revolution, we make repeated trips and the degrees we travel doesn’t just start over at 0.
We end up with a graph that looks like this:

learning coterminal angles

So what does that have to do with coterminal angles? It’s pretty easy once we think about it. Say we traveled 45° on the initial go around.

central angles

Where we started is the Initial side of the angle. Where we stopped is the terminal side of the angle. Neat, huh?
If we were going in the opposite direction of the initial side, we would be going in a negative direction. In this case, starting at the initial side and going “backwards” to the terminal side gives us -315°. When we make a complete loop, degrees traveled will be 360° and the negative angle will be 0°. Makes sense, right?
But what if we kept going after we completed the loop and landed back at the terminal side? It would look like this:

calculating coterminal angles

See how we traveled from in a complete loop from the terminal side?
To find out how many degrees we traveled in, simply add 360° to the initial angle!
We can say that 45° and 405° are coterminal.
But we can also do more! Coterminals can be negative as well. Remember the -315° from going backwards? That angle also shares the same initial and terminal sides.
Thus, 45°,-315°,and 405° are all coterminals!
*****BASIC RULE: just add or subtract 360° to your initial angle and you can find a coterminal. The possibilities are endless! Take a gander below:

learning coterminal angles online

63° and 2223° are coterminals. And we can keep going!
How else can we measure coterminals? Well, we don’t have to use degrees. We can use radians.
What’s a radian?
A radian is the measurement of an angle in a circle where the radius is the same as the angle’s arc.
Here’s a radian:

what is radian

How many radians are in a circle? Well, the formula we use is: there are 2π radians in a complete circle. We divide it like so:


Have a circle is π, and a complete circle would bring us back to 2π.
How much is 2π?
We know that π=3.14159….and then goes on forever and ever and ever.
That makes 2π=6.28318
Since there are 2π in a complete circle, that means there are 6.28318 radians in a circle.
That means that 6.28318 radians=360°
One rad is 57.296°!
Awesomely Rad, I know!
The same principle applies to radians as they do degrees.
If we need to find a coterminalangle, we can add or subtract 2π!
It’s as easy as that.
Quick example: we have an angle with radian π/3. What is the coterminal angle?

finding coterminal angle with a given radian

Just add 2π!
π/3+2π→ 2π/6+12π/6→14π/6→7π/3 radians
If we want to find more coterminal angles, we can add or subtract 2π! The possibilities are endless!
Did you want to convert this measurement to degrees?
7π/3 × (360°)/2π→7/1×120/2→7×60=420°
Here are the basic formulas:
For coterminals of degrees -> add or subtract 360°
For coterminals of radians -> add or subtract 2π
Here’s a quick cheat sheet conversion chart from radians to angles:

radians to angles

Congrats! You are now a radical coterminal angle wizard with the help of online learning StudyGate.com platform !

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How to Determine a Reference Angle

acute angle and obtuse angle

Here’s a secret that math tutors keep to themselves: Sometimes, the fanciest math skills are actually the simplest ones! Like the ultra-mysterious Reference Angle! Sounds pretty impressive, no?

But Reference Angles are actually one of the easiest things to define. Let’s draw a graph and an acute angle (an angle less than 90) and an obtuse angle (an angle that is greater than 90):

acute angle and obtuse angle

The Reference Angles are the measurement of degrees from the shortest distance between the terminal line to the X-axis.

For an acute angle, it is very simple. It is just the degrees.Inn  this case, it is 45°.

reference angle

Fort an obtuse angle, it is a little trickier. The terminal line is actually closest to the x-axis on the OPPOSITE side of the angle. In this case, it would be this:

calculating reference angle of an obtuse angle

We know the degree measurement is 70° because the angles that make up a straight line through the y-axis is 180°. We just subtract 110° from 180° to get the REFERENCE ANGLE!

Easy, huh?

***Please note that even though the Reference Angle is on the negative side of the x-axis, it is always positive.

***Also, unless the terminal side of the angle is on the y-axis to form a right angle, the Reference Angle will always be acute!!!

In other words, Reference Angle 90°

Here is the largest Reference Angle possible:

largest reference angle

It is right angle with a measurement of 90°

***We can use Reference Angles to calculate the functions of angles, like sine, cosine, and tangents. It’s basically a cool shortcut: the sin(70) and sin(110) are the same because they have the same Reference Angle!

Want to see Reference Angles in action??? Check out the link below:


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How to Prove There are 2 Similar Triangles

Ways to prove there are 2 similar triangles

In the whole, wide world of geometry, there are so many shapes and sizes of triangles that you might think they’re all exactly the same, or that they’re all completely different. Not necessarily! Two triangles must share precisely specific traits in order to be deemed similar. If you’re faced with the puzzle of determining if there are two similar triangles, there are three different paths you can take to solve the mystery.

Path 1: Use the Angle-Angle (AA) Method

If you can determine the angle measurement of at least two angles from each triangle, the AA method is your best chance of solving the similarity mystery. If two angles of one triangle are the exact same measurement of the same two angles of the other, the two triangles must be similar. It’s really just that simple!

Path 2: Use the Side-Side-Side (SSS) Method

The SSS method is most helpful when you are given, or can determine, the length of all three sides of both triangles. If all sides on one triangle are specifically proportional to the corresponding sides of the other, then those two triangles must be similar. For example, if one triangle has a hypotenuse of 3 inches, and the second has a hypotenuse of 6 inches, those two sides are proportional because 6 is exactly twice the length of 3. If the other corresponding sides of the triangle are proportional, the triangles are similar and the mystery is solved.

Path 3: Use the Side-Angle-Side (SAS) Method

When you’re faced with two triangles that have at least one given angle measurement, and two given side measurements, the SAS method is your best bet. If the corresponding angles of the triangles have the exact same measurements, and the corresponding sides are proportional to one another, then those two triangles must be similar. Just remember that the measured angle from one triangle must be in exactly the same place as the measured angle from the other, and the same principle applies to the measurements of the two sides as well.

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Graphing Functions: A How to Guide for Graph Calculator

Drawing Graph

Math brings out the artist in you!

While the industrious student can simply whip out a graph calculator to draw up graphing functions, figuring out functions the old-fashioned way can be pretty keen. With nothing but a pencil and a piece of paper divided into little squares, we can create lines and make ‘em dance to the curve of a slope intercept form.

We all knew math was a cute little number, but here is where we prove it!

Let’s take a simple equation:

y = mx + b

This is a pretty basic format for graphing. We have a Y axis that has a direct relationship with the X axis. The ratio between Y and X is determined by M.

Let’s throw in some numbers and see where they land:

y = 2x + 6

This is as easy as it gets. Let’s whip up a graph:

Learning to Draw Graph

Let’s start easy, with x = 0.

So we know that when X is at 0, Y is 6. The “2” is the ratio between X and Y. If X was 1, then simply multiply X x 2 + 6 = 8.

We can create a quick chart between X and Y. Don’t forget that we can go into negative values for X and Y s as well!


Then, plot the points and connect to form a line!

graphing functions online learning

Pretty neat, huh?

****QUICK TIP: go back to the equation

We can break M down to represent the change over Y over the change over X. If it were a fraction, it would look like this: change over Y ÷ change over X

Since 2 is basically 2÷1, we can simply say that for every change in X, we move Y up 2 spaces! Look at the chart and do the math. See how it works?

What is M was a fraction?It still works out:

This means that for every X value we have, Y goes up by . The chart would look like:


graph calculator online learningGraphs with fraction M look less steep than graphs where M is a whole number.

 But where does this get us with graphing functions?

It’s really easy: it’s basically the same thing!

ƒ(x) = x2 + 2

In this case, Y = F(x). Since the equation is squared, there will be some curves as the Y intercept changes from positive to negative. But the procedure is the same.

So let’s throw some X values in and see what we get!


learning to draw graph in math

As you can see, we have a curve!

Imagine the possibilities: with numbers, you can draw anything!

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Why Do We Use Negative Exponents?

Why do we use negative exponents in Math

When you first see a negative exponent such as 2-2, it can appear confusing. Although you know that 2 squared is 4, what is 2 squared to the -2 power?

The answer is easier than it first appears, and will become simple to understand when you realize that a negative exponent is just a way of telling you to flip the number to become a fraction. Thus, 2-2 is a more streamlined way to tell you the real problem is 1/(22). In other words, the answer to 2-2 is 1/4.

Let’s consider another problem featuring a negative exponent, but with a twist. What would we do if we had 23/2-2?

To make it easy, let’s look at the problem as two different parts: 23 multiplied by 1/2-2. The first part, 2 to the third power, is 2 multiplied by 2 multiplied by 2, or 8. The second part featuring a negative exponent is a bit trickier.

Learning Math

We already know that 2-2 is the equivalent of 1/4, but what happens when a fraction appears as the denominator of another fraction? If you recall from other mathematics assignments, you must reverse the bases and multiply the top number with the bottom number. Therefore, we should reverse the 1/4 to 4/1, or 4. Then, we can multiply it by the top number, which is 1. In other words, 4 multiplied by 1 equals 4.

At this point, we can now put the two parts back together again and solve 23/2-2 as such:

23/2-2 = 8/2-2 = 8 x 4 = 32.

In the world of mathematics, teachers, students, and scientists regularly rely on these types of shortcuts. Negative exponents aren’t meant to complicate matters, but to give you a different way to reorganize your problems. As you become more familiar with solving negative exponent problems, you will find that they are handy and apt in many situations.

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