The unit circle is an essential tool for understanding advanced topics in mathematics such as trigonometry. Learning how to use the unit circle starts with understanding its degrees and how they relate to radians. In this article, we’ll break down all the key concepts so that you can quickly and easily figure out how to use the unit circle.
Degrees and Radians Explained
Most of us are familiar with degrees as a way to measure an angle’s size. One degree is equal to 1/360th of a circle. Radians, on the other hand, are a much less commonly used system of measuring angles. They can be thought of as the number of units it takes to “fill” up a circle. One radian is equal to 1/2π of a circle, which translates to roughly 57.3 degrees.
Radians are often used in mathematics and physics, as they are a more precise way of measuring angles. They are also used in trigonometry, as they are easier to work with than degrees. Radians are also used in computer programming, as they are more efficient than degrees when it comes to calculating angles.
Unit Circle Definitions
Now let’s take a closer look at the unit circle. In a unit circle, the radius of the circle is always equal to one. This means that all of the angles in the unit circle are based off of this one standard radius, making it easier to compare angles across different circles. The x-axis of the unit circle is made up of the points where the angle that is measured is zero. The y-axis is made up of the points where the angle is either 90 or 270 degrees.
The Relationship Between Degrees and Radians
It’s important to note that there is a direct relationship between the degrees and radians measured in a unit circle. Every 1 degree is equal to π/180 radians and every 1 radian is equal to 57.3 or 180/π degrees. This makes it easier to convert between degrees and radians when using a unit circle.
Using the Unit Circle to Calculate Angles
The most common use of the unit circle is to calculate angles. It is possible to calculate angles both in degrees and in radians using the unit circle. To do so, start by drawing a line of length one from the origin of the circle, then rotate it to the desired point in the circle. From there, you can use either trigonometric ratios or your knowledge of the properties of a circle to calculate the angle.
How the Unit Circle is Used in Trigonometry
Trigonometry is one of the best reasons for knowing how to use the unit circle. Trigonometry is a branch of mathematics that deals with the relationships between angles and sides of triangles. It relies heavily on the unit circle as a way to calculate angles and length of sides in a triangle. In fact, without the unit circle, trigonometry would be a lot harder and much less precise.
Understanding the Quadrants of the Unit Circle
The unit circle is divided into four quadrants: the first, second, third, and fourth quadrants. Allow us to explain each one in more detail: the first quadrant measures angles between zero and 90 degrees; the second quadrant measures angles between 90 and 180 degrees; the third quadrant measures angles between 180 and 270 degrees; and finally, the fourth quadrant measures angles between 270 and 360 degrees.
Memorizing Degrees for the Unit Circle
If you need to memorize some key degrees for common trigonometric ratios such as sin, cos, and tan, then try using the acronym SOHCAHTOA (which stands for Sine = opposite ÷ hypotenuse, Cosine = adjacent ÷ hypotenuse, Tangent = opposite ÷ adjacent). This will help you remember which angle has which degree measurements in each quadrant of the unit circle.
Working With Right Triangles and the Unit Circle
As previously mentioned, using right triangles together with knowledge of the unit circle can be used to calculate some very useful equations in trigonometry. For example, if you’re given two sides of a triangle and asked to calculate an angle, then all you need to do is draw a triangle on your unit circle and use trigonometric ratios to work out the answer.
Examples Showing How to Use the Unit Circle
For clear examples on how to use the unit circle, we highly recommend checking out some online tutorials or videos that show examples step-by-step. These can be very helpful for learning how to use the unit circle for common equations in trigonometry such as sine, cosine, and tangent. By looking at and working through examples, you’ll gain a much better understanding of how to use a unit circle correctly.
This article has explained how to understand and use the degrees of a unit circle. With the help of this article, you should now be well on your way to mastering trigonometry using the unit circle!
