homework 4 the unit circle

Unit Circle Calculator

What is a unit circle, unit circle: sine and cosine, unit circle tangent & other trig functions, unit circle chart – unit circle in radians and degrees, how to memorize unit circle.

Welcome to the unit circle calculator ⭕. Our tool will help you determine the coordinates of any point on the unit circle. Just enter the angle ∡, and we'll show you sine and cosine of your angle .

If you're not sure what a unit circle is , scroll down, and you'll find the answer. The unit circle chart and an explanation on how to find unit circle tangent , sine, and cosine are also here, so don't wait any longer – read on in this fundamental trigonometry calculator !

A unit circle is a circle with a radius of 1 (unit radius). In most cases, it is centered at the point ( 0 , 0 ) (0,0) ( 0 , 0 ) , the origin of the coordinate system.

The unit circle is a really useful concept when learning trigonometry and angle conversion.

Now that you know what a unit circle is, let's proceed to the relations in the unit circle.

OK, so why is the unit circle so useful in trigonometry?

Unit circle relations for sine and cosine:

• Sine is the y-coordinate ; and
• Cosine is the x-coordinate

🙋 Do you need an introduction to sine and cosine? Visit our sine calculator and cosine calculator !

Standard explanation :

Let's take any point A on the unit circle's circumference.

• The coordinates of this point are x x x and y y y . As it's a unit circle, the radius r is equal to 1 1 1 (a distance between point P P P and the center of the circle).

• By projecting the radius onto the x and y axes, we'll get a right triangle, where ∣ x ∣ |x| ∣ x ∣ and ∣ y ∣ |y| ∣ y ∣ are the lengths of the legs, and the hypotenuse is equal to 1 1 1 .

• As in every right triangle, you can determine the values of the trigonometric functions by finding the side ratios:

So, in other words, sine is the y-coordinate:

And cosine is the x-coordinate.

The equation of the unit circle, coming directly from the Pythagorean theorem, looks as follows:

Or, analogically:

🙋 For an in-depth analysis, we created the tangent calculator !

This intimate connection between trigonometry and triangles can't be more surprising! Find more about those important concepts at Omni's right triangle calculator .

You can find the unit circle tangent value directly if you remember the tangent definition:

The ratio of the opposite and adjacent sides to an angle in a right-angled triangle.

As we learned from the previous paragraph , sin ⁡ ( α ) = y \sin(\alpha) = y sin ( α ) = y and cos ⁡ ( α ) = x \cos(\alpha) = x cos ( α ) = x , so:

We can also define the tangent of the angle as its sine divided by its cosine:

Which, of course, will give us the same result.

Another method is using our unit circle calculator, of course. 😁

But what if you're not satisfied with just this value, and you'd like to actually to see that tangent value on your unit circle ?

It is a bit more tricky than determining sine and cosine – which are simply the coordinates. There are two ways to show unit circle tangent:

• Create a tangent line at point A A A .
• It will intersect the x-axis in point B B B .
• The length of the A B ˉ \bar{AB} A B ˉ segment is the tangent value

• Draw a line x = 1 x = 1 x = 1 .
• Extend the line containing the radius.
• Name the intersection of these two lines as point C C C .
• The tangent, tan ⁡ ( α ) \tan(\alpha) tan ( α ) , is the y-coordinate of the point C C C .

In both methods, we've created right triangles with their adjacent side equal to 1 😎

Sine, cosine, and tangent are not the only functions you can construct on the unit circle. Apart from the tangent cofunction – cotangent – you can also present other less known functions, e.g., secant, cosecant, and archaic versine:

The unit circle concept is very important because you can use it to find the sine and cosine of any angle. We present some commonly encountered angles in the unit circle chart below:

As an example – how to determine sin ⁡ ( 150 ° ) \sin(150\degree) sin ( 150° ) ?

• Search for the angle 150 ° 150\degree 150° .
• As we learned before – sine is a y-coordinate, so we take the second coordinate from the corresponding point on the unit circle:

Alternatively, enter the angle 150° into our unit circle calculator. We'll show you the sin ⁡ ( 150 ° ) \sin(150\degree) sin ( 150° ) value of your y-coordinate, as well as the cosine, tangent, and unit circle chart.

Well, it depends what you want to memorize 🙃 There are two things to remember when it comes to the unit circle:

Angle conversion , so how to change between an angle in degrees and one in terms of π \pi π (unit circle radians); and

The trigonometric functions of the popular angles.

Let's start with the easier first part. The most important angles are those that you'll use all the time:

• 30 ° = π / 6 30\degree = \pi/6 30° = π /6 ;
• 45 ° = π / 4 45\degree = \pi/4 45° = π /4 ;
• 60 ° = π / 3 60\degree = \pi/3 60° = π /3 ;
• 90 ° = π / 2 90\degree = \pi/2 90° = π /2 ; and
• Full angle, 360 ° = 2 π 360\degree = 2\pi 360° = 2 π .

As these angles are very common, try to learn them by heart ❤️. For any other angle, you can use the formula for angle conversion :

Conversion of the unit circle's radians to degrees shouldn't be a problem anymore! 💪

The other part – remembering the whole unit circle chart, with sine and cosine values – is a slightly longer process. We won't describe it here, but feel free to check out 3 essential tips on how to remember the unit circle or this WikiHow page . If you prefer watching videos 🖥️ to reading 📘, watch one of these two videos explaining how to memorize the unit circle:

• A Trick to Remember Values on The Unit Circle ; and
• How to memorize unit circle in minutes!!

Also, this table with commonly used angles might come in handy:

And if any methods fail, feel free to use our unit circle calculator – it's here for you, forever ❤️ Hopefully, playing with the tool will help you understand and memorize the unit circle values!

What is tan 30 using the unit circle?

tan 30° = 1/√3 . To find this answer on the unit circle, we start by finding the sin and cos values as the y-coordinate and x-coordinate, respectively: sin 30° = 1/2 and cos 30° = √3/2 . Now use the formula. Recall that tan 30° = sin 30° / cos 30° = (1/2) / (√3/2) = 1/√3 , as claimed. See how easy it is?

How do I find cosecant with the unit circle?

To determine the cosecant of θ on the unit circle:

• From the center of the circle draw the radius corresponding to the angle θ .
• Draw tangent lines to the circle at points (0,1) and (0,-1) .
• Extend the radius from Step 1 so that it intersects one of those tangents.
• The distance from the center to the intersection point from Step 3 is the cosecant of your angle θ .
• If there's no intersection point, the cosecant of θ is undefined (this happens when sin θ = 0 ).

How do I find arcsin 1/2 with the unit circle?

As the arcsine is the inverse of the sine function , finding arcsin(1/2) is equivalent to finding an angle whose sine equals 1/2 . On the unit circle, the values of sine are the y-coordinates of the points on the circle. Inspecting the unit circle, we see that the y-coordinate equals 1/2 for the angle π/6 , i.e., 30° .

Perimeter of a square

Triangle congruence.

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Calcworkshop

Unit Circle with Everything Charts, Worksheets, and 35+ Examples!

// Last Updated: January 22, 2020 - Watch Video //

The Unit Circle is probably one of the most important topics in all of Trigonometry and is foundational to understanding future concepts in Math Analysis, Calculus and beyond.

The good thing is that it’s fun and easy to learn!

Everything you need to know about the Trig Circle is in the palm of your hand.

In the video below , I’m going to show my simple techniques to quickly Memorize the Radian Measures and all Coordinates for every angle!

• No more trying to calculate all those angles.
• No more getting frustrated when asked to evaluate or memorize each and every coordinate.

Together, we are going to become human calculators, and bring our mathematical genius to life!

What is the Unit Circle?

Well, the Unit Circle, according to RegentsPrep, is a circle with a radius of one unit, centered at the origin.

Why make a circle where the radius is 1, you may ask?

Reference Triangle in the First Quadrant of the Unit Circle

But, the Unit Circle is more than just a circle with a radius of 1; it is home to some very special triangles.

Remember, those special right triangles we learned back in Geometry: 30-60-90 triangle and the 45-45-90 triangle? Don’t worry. I’ll remind you of them.

30-60-90 Triangle

45-45-90 Triangle

Well, these special right triangles help us in connecting everything we’ve learned so far about Reference Angles, Reference Triangles, and Trigonometric Functions, and puts them all together in one nice happy circle and allow us to find angles and lengths quickly.

In other words, the Unit Circle is nothing more than a circle with a bunch of Special Right Triangles.

Unit Circle with Special Right Triangles

Now, I agree that may sound scary, but the cool thing about what I’m about to show you is that you don’t have to draw triangles anymore or even have to create ratios to find side lengths.

The Unit Circle

Everything you see in the Unit Circle is created from just three Right Triangles, that we will draw in the first quadrant, and the other 12 angles are found by following a simple pattern! In fact, these three right triangles are going to be determined by counting the fingers on your left hand!

How to Memorize the Unit Circle

Ok, so there are two ways you can do this:

• Use a Unit Circle Chart
• Simply know how to count

If it were me, I’d just want to count and not have to memorize a table, and that’s what I’m going to show you.

The Unit Circle has an easy to follow pattern, and all we have to do is count and look for symmetry. Moreover, everything you need can be found on your Left Hand.

If you place your left hand, palm up, in the first quadrant your fingers mimic the special right triangles that we talked about above: 30-60-90 triangle and the 45-45-90 triangle.

I will show you how to remember each angle, in radian measure, for each of your fingers and also how to find all the other angles quickly by using the phrase:

All Students Take Calculus!

For a quick summary of this technique, you can check out my Unit Circle Worksheets below.

And after you know your Radian Measures, all we have to do is learn an amazing technique called the Left-Hand Trick that is going to enable you to find every coordinate quickly and easily.

Furthermore, this Left-Hand Trick is going to help you not only to memorize the Unit Circle, but it is also going to allow you to evaluate or find all six trig functions!

Additionally, as Khan Academy nicely states, the Unit Circle helps us to define sine, cosine and tangent functions for all real numbers , and these ratios (that we have sitting in the palm of our hand) be used even with circles bigger or smaller than a radius of 1.

Isn’t that awesome?

Yes, indeed!

As you’re watching the video, you’re going to learn how to:

• Draw the Unit Circle.
• Generate every Radian Measure just by counting.
• Use the Left-Hand Trick to find the coordinates of every angle.
• Evaluate all six trigonometric functions for each and every angle on the Unit Circle.

Unit Circle Worksheets

• Blank Unit Circle Worksheet : Practice your skills by identifying the Radian Measure, Degree Measure and Coordinate for each angle.
• How to Memorize the Unit Circle : Summary of how to remember the Radian Measures for each angle.
• Left-Hand Trick : How to find sin cos tan sec csc cot for every angle.
• Unit Circle Chart : Complete Unit Circle with all Degrees, Radian, and Coordinates.

Unit Circle Video

1 hr 38 min

• Intro to Video: Unit Circle
• 00:00:40 – Quick Review of the Six Trig Functions + How to represent them in a Trig Circle
• 00:07:32 – Special Right Triangles & their Importance
• 00:23:51 – Creating the Unit Circle + Left Hand Trick!
• 00:46:37 – Examples #1-7
• 00:55:32 – Examples #8-18
• 01:09:45 – Examples #19-27
• 01:25:35 – Examples #28-36

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Interactive Unit Circle

Sine, Cosine and Tangent ... in a Circle or on a Graph.

Sine, Cosine and Tangent

Sine , Cosine and Tangent (often shortened to sin , cos and tan ) are each a ratio of sides of a right angled triangle:

For a given angle θ each ratio stays the same no matter how big or small the triangle is

Helping math teachers bring calculus to life

The Unit Circle (Lesson 4.3 Day 1)

Unit 4 - day 4, ​learning objectives​.

Use the legs of special right triangles with hypotenuse of 1 to find ordered pairs on the unit circle at key angles.

Use reference angles to explain the symmetry of the unit circle in the four quadrants.

Quick Lesson Plan

Activity: coming full circle.

Lesson Handout

Experience first.

In numbers 1 and 2, the students will write ratios for sine and cosine when given a triangle with all of the side lengths.  We want them to see that when the hypotenuse is 1, the actual side length of the triangles are the same as the sine and cosine ratios for specific angles.  This will lead them to the unit circle, where the radius is 1 and the values of sine and cosine at different angles are equal to the x- and y-coordinates at those spots.  The idea here is that students don’t get caught up in “Sine = y and cosine = x,” but they keep up with the understanding that sine is the ratio of the opposite side to the hypotenuse and cosine is the ratio of the adjacent side to the hypotenuse.  In number three, they’ll develop the understanding of the first quadrant of the unit circle as it relates to evaluating sine, cosine, and tangent of different angles.

Formalize Later

The sequencing of questions in this lesson was developed strategically so the students would understand the concept behind “the unit circle” and how it helps us evaluate trigonometric ratios for specific angles.  Again, it’s important that students understand why the cosine is the x-coordinate and the sine is the y-coordinate, so be sure to keep referring back to the ratios from the triangles throughout the debrief.  Show them how to extend their knowledge to eagles like 0 degrees and 90 degrees so that they’ll be prepared for looking at angles greater than 180 degrees or less than 0 degrees in the next lesson.

• Nov 30, 2021

Teaching the Unit Circle WITHOUT Memorization

Updated: Aug 9

I don’t think there are any topics in high school math with more tricks and gimmicks than completing a blank unit circle. Don’t believe me? Search Unit Circle Tricks on YouTube. There are videos with millions of views, teaching students how to blindly complete a unit circle. I will admit, showing students these tricks does help them to memorize a unit circle. But what is the point of memorizing a bunch of fill-in-the-blank answers if there is no understanding of what they mean?

The good news is you don’t have to choose between understanding and accuracy. You can teach the unit circle so that students don’t have to memorize AND they develop a conceptual understanding of the unit circle. Students will be able to reason their way through identifying all the angles and coordinates BEFORE they ever see a completed unit circle.

So how are we going to make this happen? Special right triangles.

Turns out the Unit Circle is just a whole bunch of special right triangles! Did you know that? I didn’t, and it truly blew my mind when I realized it. But why does that matter? Well, with a couple of carefully chosen special triangles, you can find all angles and coordinates of the unit circle. Let’s look at how.

Spotlight Lesson:

Algebra 2 lesson 9.6: the unit circle.

It’s important to know what prerequisite knowledge students need for this lesson. In our Algebra 2 Trigonometry unit, students have just gone over special right triangles and angles in standard position on the coordinate plane in the previous two lessons. They have not learned about radians yet so we only fill in the unit circle with degrees today and will come back to fill in the radians later.

Students will start out the activity by finding sides lengths for a 30-60-90 triangle and 45-45-90 triangle that both have a hypotenuse of 1 . ( Use this unit circle and set of triangles ). Once they’ve done that, they should cut out the set of triangles that you give them and label all sides and all angles on both sides of the paper . Color coding the sides is really helpful for seeing the patterns. Students would need 4 different colors to do this.

Students will use these triangles to fill in the angles and coordinates for all of the points marked on the unit circle by fitting the angles of the triangle into the reference angle of the circle and then using the side lengths of the triangle to determine the x and y coordinates. You’ll want to model how this works for at least two of the angles, if not three. If possible, project your handout and triangle to show how you are maneuvering the triangle while you explain your thinking.

"Let’s look at this first angle. I can see that my 30-60-90 triangle will fit in here if I put the 30-degree angle at the origin. So this angle must be 30 degrees. Now I need to fill in the ordered pair, so I need to find the x and y. Well, I know that x represents the horizontal distance from the origin. So how far over from the origin have I gone? How long is this width? (Show with your fingers what length you’re talking about. Try to get students to notice that we already have this width.) Oh, well the distance over is the same as this leg of my triangle which I know is square root of 3 over 2. So the x value is square root of 3 over 2! Now what about the y? That is how high up we have gone. I can see that’s the other side of my triangle which is ½ so y is ½."

Next, I would Think Aloud through an angle that doesn’t have 30 degrees as the reference angle. I like doing the 60-degree angle because students don’t always notice that they can turn their triangle that way.

I generally like to show at least one more angle that requires reflecting the triangle, like the 135-degree angle. During your modeling, you’ll want to talk through how you found the angle by being 45 degrees short of 180 so the angle must be 135. You’ll also should mention that we are now to the left of the origin so the x value is negative.

After modeling the thinking for these examples, give students time to manipulate their triangles and find all of the missing sides and angles. By the end of the activity, they’ll not only have a beautiful unit circle, but they’ll also be able to reason their way to finding any of the angles and coordinates. No memorizing required!

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• 8.1 Representing Data
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• 12.1 Functions and Transformations
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12.4.1a Angles and Radian Measure

• Ch4 Unit Outline
• p175 #1-6, 12-15, 17, 19, C1, C2, C3

Things you should be able to do after today:

• know the relationship between radians, the radius and the arc length in a circle
• convert degrees to radian measure
• convert radians to degrees
• know the special angles in degrees and radians 

12.4.1b Coterminal Angles

Angles in standard position use the positive side of the x-axis as the inital arm.  Counter-clockwise rotations from this initial arm produce positive angles, and clockwise rotations produce negative angles.  Sometimes, different angles can end up with the same terminal arm.  These are called coterminal angles.

Assignment:

• p175 #7-9, 11, 18-20 C5

Things you should know how to do after today:

• define an angle in standard position
• define a coterminal angle
• Find the reference angle for an angle in standard position
• Find the smallest positive coterminal angle for any angle in standard position

12.4.2a The Unit Circle

The foundation for a lot of trigonometry is the unit circle.

• p187 #1-4, 10, 11
• recall the relationship between radian measure with the number of radiuses around the circumference of a circle
• using special triangles, determine the coordinates for a point on the terminal arm when given the measure of the a standard position angle in radians

12.4.2b Unit Circle

• p187 #5, 6, 7, 9, 12, 13, 15, C1 C2 *17 *18
• Given the coordinates of a point on the unit circle, determine the measure of the standard position angle in radians

12.4.3a Trigonometric Ratios

• Example 2, DEF from the the notes will be marked next class
• p201 #1-9, 13, 14, 17, 18 C3
• sinθ, cosθ, tanθ, secθ, cscθ and cotθ

12.4.3b Finding the Arc Length

• Notes: 12.4.3b
• p201 #10, 11, 12, 15, 16, C1, C2 *20,22, 24

12.4.4 Solving Trigonometric Equations

• Notes: 4.4a
• Notes: 4.4b
• p211 #1, 3, 5, 6, 10, 11, 15, 16
• specific domains
• general solutions (solved over the reals)

12.4.5 Review

Warmup Answer Key:

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