Sample Unit Circle Charts & Diagrams

What Are Unit Circle Charts & Diagrams?

A unit circle diagram is a tool used by mathematicians, teachers, and students like you to easily solve for sine, cosine, and the tangent of an angle of a triangle. This is a circle that consists of a radius that is equivalent to 1, which means that any straight line you draw from the circle’s center point up to its edge, will always be equivalent to 1. Furthermore, this circle has an x and y-axis intersecting in the middle, separating its negative and positive quadrants, serving as your guide in calculating for the cosine, sine, and tangent of any angle from 0° to 360° (also equivalent to 0 and 2? radians).

What Are the Different Trigonometric Identities?

Trigonometric identities are essential when you wish to simplify mathematical expressions related to trigonometric functions, especially with its equations valid for right-angled triangles. Some of these trigonometric identities are listed below:

Right Triangle: A right triangle, if you noticed, always has an angle of 90 degrees. Like any other triangles, the right triangle composed of three sides and three angles with a hypotenuse located on the opposite side of its right angle. Using the Pythagorean Theorem, you can solve for the missing sides of the right triangle. Pythagoras Theorem: A Pythagoras Theorem, or what you mostly know as the Pythagorean Theorem, is closely related to the right triangle since it defines the relationship between the three sides of the triangle. This theorem shows that the hypotenuse of a triangle equals the sum of the squares of its two other sides. It even follows an equation (the Pythagorean equation):

a2 + b2 = c2
Wherein, the “C” serves as your hypotenuse, while your “a” and “b” serves as your triangle’s two other sides (the opposite side and the adjacent side).Sine, Cosine, and Tangent: The sine, cosine, and tangent are considered as your three primary functions in Trigonometry. Your sine function represents the ratio of your triangle’s length of the opposite side and your hypotenuse. Your cosine function, on the other hand, represents the ratio of your triangle’s length of the adjacent side and your hypotenuse. In contrast, your tangent function represents the ratio of your triangle’s length of the opposite side and your length of the adjacent side. Cosecant, Secant, and Cotangent: Aside from the three primary functions in Trigonometry, you also have the cosecant, secant, and cotangent. In a right triangle, your cosecant of an angle is equivalent to your hypotenuse being divided by your length of the opposite side. Your secant of an angle, on the other hand, is equal to your hypotenuse being divided by the length of your adjacent side. In contrast, your cotangent of an angle is equivalent to your length of the adjacent side being divided by your length of the opposite side.

How to Apply Trigonometry in Real Life?

Trigonometry is not just a simple study of triangles, lengths, heights, and angles. It also has a lot of functions in real life. In fact, this made a significant impact on the development under the fields of Engineering, aviation, astronomy, and many more that require familiarization of sines and cosines to create an output.

How to Use a Unit Circle Diagram?

There are different ways to solve for cosine, sine, and tangent of a right triangle. You can solve it using a scientific calculator, or you may use the Pythagorean equation. However, what if you don’t have your calculator with you? How are you going to answer your exam? Well, there is another simpler way to do it. If you are studying trigonometry or calculus, how familiar are you with the unit circle? A unit circle can help you solve for the sine, cosine, and tangent of an angle by merely familiarizing yourself with its different coordinates. Find out more about how to use the unit circle and read these few tips below.

Step 1: Familiarize Yourself with the Major Angles and Radians of the Unit Circle

The unit circle is composed of radians, degrees, and major angles. The first step is to familiarize yourself with them. A circle is equivalent to a total of 360 degrees and is divided into four quadrants, wherein each quadrant has a total value of 90 degrees and is further divided into 30 degrees, 45 degrees, and 60 degrees each. Make sure to remember them together with their radian equivalents as they will come in handy later.

Step 2: Identify the Different Quadrants of the Unit Circle

As stated earlier, a unit circle is divided into four quadrants. You have quadrant one, located at the right upper side of the circle, and your quadrant two located at the upper left side of the circle. On the other hand, your quadrant three is located on the lower left side, while your quadrant four is located at the lower right side of the circle. Each of these quadrants is created with your x and y-axis intersecting in the middle of the circle. Identify the different quadrants of the unit circle and keep in mind that each quadrant has an equivalent positive and negative value. 

Step 3: Familiarize Yourself with the Coordinates of Point on Circle

Now that you know the different radians, degrees, angles, and quadrants that make up the unit circle, the next is to familiarize yourself with the various coordinates of points on the circle. Each of these coordinates has corresponding angle degrees and radians equivalent. Make sure to determine which quadrant has a positive and negative x and y-axis since your x-coordinates are equivalent to your cosine while your y-coordinates are equivalent to your sine.

Step 4: Determine and Solve for Your Tangent

Lastly, now that you are familiar with the different parts of the unit circle, it’s time to solve for your tangent. Since your unit circle does not have a tangent, you need to use a formula to solve it. With your cosine and sine readily available in your unit circle, finding your tangent won’t take long. Simply use the method: tan? = sin? / cos? Keep these tips in mind, and you shall have no problem acing your math tests in no time.

FAQs

Where did the unit circle come from?

The unit circle, as well as Trigonometry, has been used by Egyptian mathematicians to build pyramids and Babylonian astronomers to keep track of stars and planets’ motion way back from the 2nd millennium BC. Later in history, Greek and Hellenistic mathematicians made use of the table of chords produced by Hipparchus, making him the father of Trigonometry. Soon, these were further developed and translated by various mathematicians around the world over time, until it was molded into the Trigonometry and unit circle from what you know it today.

How many quadrants does a unit circle have?

A unit circle is composed of four quadrants, namely, Quadrant I (QI), Quadrant II (Q II), Quadrant III (Q III), and Quadrant IV (Q IV). Quadrant one is composed of Both positive x and y coordinates (+X,+Y), Quadrant two is composed of negative and positive coordinates (-X,+Y). Quadrant three, on the other hand, is composed of both negative coordinates (-X,-X), while Quadrant four is composed of positive and negative coordinates (+X,-Y).

What is a reference angle?

A reference angle is a positive acute angle you find inside a unit circle that represents an angle of any measure. If you notice, this angle remains between 0 and 90 degrees, which makes it the smallest angle you find from the terminal side (where the angle ends) of the angle with the x-axis.

Can a reference angle be negative?

A quick answer is no, reference angle is always positive and is always less than or equal to 90 degrees, even for large angles regardless of which quadrant you place them since they always measure between 0 and 2.

Why is the unit circle important in Trigonometry?

Trigonometry is the study of triangles. Since the unit circle serves as a tool in helping you identify and calculate for the different trigonometric functions, which is one of the building blocks of the subject, it became essential in Trigonometry.

Now that you know what a unit circle is and what it’s used for, you are probably wondering what’s the point of learning all these cosines, sines, and tangents. If you don’t know, Trigonometry is a part of your life, especially when you work under the fields of engineering, astronomy, architecture, aviation, and criminology. When you use the unit circle to solve in Trigonometry, you are not merely solving for a missing side or angle of a triangle, but you are trying to identify and keep track of the trigonometry ratios and their relationships with one another. Trigonometry is never an easy subject, especially when you are still a beginner. Nevertheless, do not overthink and just have fun solving!