unit 12 trigonometry 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° .

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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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Course: Algebra 2   >   Unit 11

• Unit circle
• The trig functions & right triangle trig ratios
• Trig unit circle review

Chapter 6: Radians

Exercises 6.2 The Circular Functions

• Know the trigonometric function values for the special angles in radians #1–4, 46–48
• Use a unit circle to find trig values #5–30, 45–58
• Find reference angles in radians #33–45
• Evaluate trigonometric expressions #31–32, 49–54
• Find coordinates on a unit circle #55–60, 67–68
• Find an angle with a given terminal point on a unit circle #61–66
• Use the tangent ratio to find slope #69–74
• Find coordinates on a circle of radius #77–80

Suggested Problems

Problems: #2, 6, 14, 18, 22, 30, 54, 33, 42, 32, 60, 66, 70, 72, 78

Exercises Homework 6.2

Exercise group.

For Problems 1–4, each point on the unit circle is the terminal point of an angle in standard position. Give exact values for the radian measure, [latex]t{,}[/latex] of the angle and the coordinates [latex](x,y)[/latex] of the point.

For Problems 5–8, use the unit circle to estimate the sine, cosine, and tangent of each arc of given length.

• [latex]0.4[/latex]
• [latex]1.2[/latex]
• [latex]2[/latex]
• [latex]0.8[/latex]
• [latex]2.6[/latex]
• [latex]4[/latex]
• [latex]2.8[/latex]
• [latex]3.5[/latex]
• [latex]5[/latex]
• [latex]3[/latex]
• [latex]4.3[/latex]
• [latex]5.5[/latex]

For Problems 9–14, use the unit circle to estimate two numbers with the given trig value.

[latex]\cos t = 0.3[/latex]

[latex]\sin t = 0.1[/latex]

[latex]\sin t = -0.7[/latex]

[latex]\cos t = -0.6[/latex]

[latex]\tan t = \dfrac{-4}{9}[/latex]

[latex]\tan t = \dfrac{8}{6}[/latex]

Each of Problems 13–20 describes an arc in standard position on the unit circle. In which quadrant does the terminal point of the arc lie?

[latex]\sin s \gt 0,~ \cos s \lt 0[/latex]

[latex]\sin s \lt 0,~ \cos s \gt 0[/latex]

[latex]\cos t \lt 0,~ \tan t \lt 0[/latex]

[latex]\cos t \gt 0,~ \tan t \lt 0[/latex]

[latex]\sin x \lt 0,~ \tan x \gt 0[/latex]

[latex]\sin x \gt 0,~ \tan x \lt 0[/latex]

For Problems 21–26, without using a calculator, decide whether the quantity is positive or negative.

[latex]\cos 2.7[/latex]

[latex]\sin 4.1[/latex]

[latex]\tan 3.8[/latex]

[latex]\tan 5.4[/latex]

[latex]\sin 2.2[/latex]

[latex]\cos 4.9[/latex]

For Problems 27–30, place the trig values in order from smallest to largest. Use the figure to help you, but try not to use a calculator!

[latex]\sin 0.5,~ \sin 1.5,~ \sin 2.5,~ \sin 3.5[/latex]

[latex]\cos 1.6,~ \cos 2.6,~ \cos 3.6,~ \cos 5.6[/latex]

[latex]\cos 2,~ \cos 3,~ \cos 4,~ \cos 5[/latex]

[latex]\sin 2.8,~ \sin 3.8,~ \sin 4.8,~ \sin 5.8[/latex]

The sunrise time in Wellington, New Zealand, on the [latex]n[/latex]th day of the year can be modeled by [latex]S = 1.93 \sin (0.016n - 1.13) + 6.14[/latex] where [latex]S[/latex] is given in hours after midnight. Find the sunrise time on January 1 (day [latex]n = 1[/latex]), April 1 (day [latex]n = 91[/latex]), July 1 (day [latex]n = 182[/latex]), and October 1 (day [latex]n = 274[/latex]).

The variable star RT Cygni reached its maximum magnitude on May 22, 2004, and [latex]t[/latex] days later, its magnitude is given by [latex]M = 9.55 - 2.25 \cos (0.033t)[/latex] Find the magnitude of RT Cygni on days [latex]t = 0,~ t = 48,~ t = 95,~ t = 142,[/latex] and [latex]t = 190{.}[/latex] (Note that smaller values of [latex]M[/latex] denote brighter magnitudes.)

For Problems 33–38, find the reference angle in radians, rounded to two decimal places.

[latex]1.8[/latex]

[latex]4.9[/latex]

[latex]-2.3[/latex]

[latex]-6.0[/latex]

[latex]9.4[/latex]

[latex]7.1[/latex]

For Problems 39–44, find the reference angle in radians, expressed as a multiple of [latex]\pi{.}[/latex]

[latex]\dfrac{11\pi}{12}[/latex]

[latex]\dfrac{11\pi}{8}[/latex]

[latex]\dfrac{4\pi}{3}[/latex]

[latex]\dfrac{7\pi}{6}[/latex]

[latex]\dfrac{13\pi}{4}[/latex]

[latex]\dfrac{8\pi}{3}[/latex]

Find three angles in radians between [latex]0[/latex] and [latex]2\pi[/latex] with the given reference angle. Sketch all the angles on a unit circle.

• [latex]\dfrac{\pi}{6}[/latex]
• [latex]\dfrac{\pi}{4}[/latex]
• [latex]\dfrac{\pi}{3}[/latex]

Complete the table.

For Problems 49–54, evaluate the expression exactly.

[latex]\cos \dfrac{\pi}{3} \sin \dfrac{\pi}{6}[/latex]

[latex]\sin \dfrac{\pi}{4} \tan \dfrac{\pi}{3}[/latex]

[latex]\tan \dfrac{5\pi}{6} + \tan \dfrac{7\pi}{4}[/latex]

[latex]\cos \dfrac{3\pi}{4} - \cos \dfrac{5\pi}{3}[/latex]

[latex]\cos^2 (\dfrac{11\pi}{6}) - 3\cos \dfrac{11\pi}{6}[/latex]

[latex]2\sin \dfrac{4\pi}{3} - \sin^2 (\dfrac{4\pi}{3})[/latex]

Starting at [latex](1,0){,}[/latex] you walk [latex]s[/latex] units around a unit circle. For Problems 55–58, sketch a unit circle showing your position. What are your coordinates?

[latex]s = 2.5[/latex]

[latex]s = 4.3[/latex]

[latex]s = 8.5[/latex]

[latex]s = 11[/latex]

City Park features a circular jogging track of radius 1 mile, centered on the open-air bandstand. You start jogging on the track 1 mile due east of the bandstand and proceed counterclockwise. What are your coordinates, relative to the bandstand, when you have jogged five miles?

Silver Reservoir is a circular man-made lake of radius 1 kilometer. If you start at the easternmost point on the reservoir and walk counterclockwise for 4 kilometers, how far south of your initial position are you?

For Problems 61–66, find the angle in radians between [latex]0[/latex] and [latex]2\pi[/latex] determined by the terminal point on the unit circle. Round your answer to hundredths.

[latex](-0.1782, 0.9840)[/latex]

[latex](-0.8968, -0.4425)[/latex]

[latex](0.8855, -0.4646)[/latex]

[latex](0.9801, 0.1987)[/latex]

[latex](-0.7659, -0.6430)[/latex]

[latex](0.9602, -0.2794)[/latex]

• Sketch a unit circle and the line [latex]y = x{.}[/latex] Find the coordinates of the two points where the line and the circle intersect.
• State your answers to part (a) using trigonometric functions.
• Sketch a unit circle and the line [latex]y = -x{.}[/latex] Find the coordinates of the two points where the line and the circle intersect.
• Sketch a line that passes through the origin and the point [latex](8,3){.}[/latex] What is the slope of the line?
• What is the angle of inclination of the line in radians, measured from the positive [latex]x[/latex]-axis?
• Sketch a line that passes through the origin and the point [latex](3,8){.}[/latex] What is the slope of the line?

For Problems 71–74, find an equation for the line with the given angle of inclination, passing through the given point. (See Section 4.3 to review angle of inclination.)

[latex]\alpha = \dfrac{\pi}{3},~ (4,2)[/latex]

[latex]\alpha = \dfrac{5\pi}{6},~ (-6,3)[/latex]

[latex]\alpha = 2.4,~ (5,-8)[/latex]

[latex]\alpha = 0.6,~ (-2,-3)[/latex]

Use similar triangles to show that the coordinates of point [latex]P[/latex] on the unit circle shown at right are [latex](\cos t, \sin t){.}[/latex]

Use similar triangles to show that [latex]ST = \tan t{.}[/latex]

Use the results of Problem 77 for Problems 78–80.

Use similar triangles to show that the coordinates of a point [latex]P[/latex] determined by angle [latex]\theta[/latex] on a circle of radius [latex]r[/latex] are [latex]x = r \cos \theta,~ y = r\sin \theta{.}[/latex] (See the figure at right.)

The Astrodome in Houston has a diameter of 710 feet. If you start at the easternmost point and walk counterclockwise around its perimeter for a distance of 250 feet, how far north of your starting point are you?

The Barringer meteor crater near Winslow, Arizona, is 1182 meters in diameter. You start at the easternmost point on the rim of the crater and walk counterclockwise around the edge. After walking for 1 kilometer, what is your position relative to your starting point?

One of the most intriguing features of Stonehenge is the position of the four Station Stones.

They form the corners of a rectangle inscribed in the Aubrey Circle on the perimeter of the henge, which has diameter 288 feet. A line from the center of the circle and perpendicular to the long edge of the rectangle points through the Slaughter Stones at the entrance of the henge and out to the Heel Stone. If you stood in the center of the circle on the summer solstice, you would see the sun rise directly over the Heel Stone.

• The sun rises [latex]48.6°[/latex] east of north on the summer solstice at Stonehenge. If the positive [latex]y[/latex]-axis points north, find the coordinates of the henge entrance relative to its center.
• The northernmost station stone is located [latex]66.6°[/latex] of arc counterclockwise from the entrance. Find its coordinates relative to the center of the henge.

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

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7.3E: Unit Circle (Exercises)

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30. Find the exact value of \(\sin \frac{\pi}{3}\).

31. Find the exact value of \(\cos \frac{\pi}{4}\)

32. Find the exact value of \(\cos \pi\).

33. State the reference angle for \(300^{\circ}\).

34. State the reference angle for \(\frac{3 \pi}{4}\).

35. Compute cosine of \(330^{\circ}\).

36. Compute sine of \(\frac{5 \pi}{4}\).

37. State the domain of the sine and cosine functions.

38. State the range of the sine and cosine functions.

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