Are you enrolled in or going to enroll in a calculus/trigonometry class? If you answered yes, it’s a good idea to learn about and use the unit circle. That is one of the most significant mathematical tools for calculating the cosine, sine, or tangent of an angle quickly and simply. How does the unit circle, on the other hand, work?
Trigonometric ratios are defined inside the boundaries of right angles. The sum of a triangle’s three interior angles (right triangle) is always 180 degrees. Furthermore, because one of the angles is 90 degrees, the other two must be acute. It is possible to define the cosine, sine, and tangent ratios for any angle using this knowledge about the right triangle.
The simplest way to solve for these ratios is to use the unit circle. We’ll learn more about the unit circle in this tutorial, including what it is, how it works, how to memorize it, and how to use it to solve trig issues.
A Closer Examine Of The Unit Circle
Any circle with a radius of one is referred to as a unit circle (trig circle). That also implies that the circle’s diameter is two (diameter is always double the length of the radius). The point where the y-axis and the x-axis intersect is the center. See the illustration below.
Figure 1: The radius and right triangle of a unit circle are shown.
The unit circle allows you to quickly compute the sine, cosine, or tangent of any angle between 0 and 360 degrees. Drawing a radius at any angle creates a right triangle, as seen in the picture above (Figure 1).
If you look closely at the triangle, you’ll see that sine is the y-coordinate (vertical line) and cosine is the x-coordinate (horizontal line) (horizontal line). The radius of the triangle also becomes the hypotenuse.
We now know that the Pythagoras theorem () may be used to solve for the lengths of any side of a triangle (where a and b represent the lengths of the triangle while c is the hypotenuse). The formula for right triangles in a unit circle may be simply deduced from the following equation.
We already know that we can simplify our equation by writing it like this: In the unit circle, we can now compute the principal angles in degrees and radians. Take a peek at:
Figure ii: Angles in degrees and radians are shown in a unit circle.
“What about when you just have a straight line (no triangle is formed)?” is a question you may wish to pose at this point. The y-coordinate (sin 0°) will be zero, and the x-coordinate (cos 0°) will be one in this situation.
Why Should You Understand How To Use The Unit Circle?
“How is this important?” you may question as you examine the application of the unit circle. There are many reasons why you should understand the concept:
- The unit circle has the benefit of allowing you to solve for the cosine, tangent, and cosine of any radian or degree. This is critical, particularly in your math and physics subjects. It’ll also help you in practical sciences like geography and chemistry. As a result, you’ll need to know it to pass most topics that need some kind of trigonometry, either directly or indirectly.
- The unit circle makes studying mathematics easier by eliminating the need to recall a large number of topics. In most situations, students must learn over 15 angles and their values in order to calculate the cosine, sine, and tangent of angles accurately. When utilizing the unit circle, however, students do not need to remember all of them. All they have to do now is think about how they’ll use it.
- Trigonometry, unlike other topics, is a vast subject. Construction, aerodynamics, shooting, and engineering are all professions that require it in real life. The unit circle is a crucial lever for student achievement because it makes trigonometry and mathematics simple to learn.
How To Use The Unit Circle – Trigonometry Circle Problem Solving
It is necessary to examine the unit circle chart more closely in order to properly apply the unit circle to trig issues.
A comprehensive unit circle chart is shown in Figure iii.
You will find calculus to be a reasonably straightforward topic if you grasp how the chart works before beginning the course. Take a look at the issue below to see how it works. You will not be permitted to use a calculator in the majority of circumstances.
- To solve, use the following formula: Sin 30°
There is no need to get overwhelmed in order to address the situation. Just return to the unit circle. At 30°, you’ll see that the y-coordinate value is 12. We can simply compute the solution since the y-coordinate = sine: 30° Sin = 1/2
Calculate the cosine (x-coordinate) for the issue using the unit circle. It is in this scenario. But what would the response be like? Take a look at this:
It’s really rather simple!
Tips For Remembering And Using The Unit Circle Trig
When solving a trig issue, the unit circle is without a doubt the simplest and most easy way to use. But hang in there! The chart in Figure iii will not give you with any information.
So, what’s the best way to remember the unit circle chart?
Here are some helpful hints to help you remember the unit circle chart and use it correctly:
- Pay Special Attention To The Common Angles And Their Coordinates: Make sure you have the usual angles at your fingertips while using the unit circle. This requires a thorough understanding of their radians, degrees, and coordinates. Once you’ve figured out the common coordinates, it’ll be much easier to figure out the angles that lie between them.
- Understand What Is Negative And Positive: Knowing which y-coordinates and x-coordinates are negative is critical to solving any trigonometric issue. You must understand at this stage that whether a coordinate is positive or negative is determined by which quadrant it falls into. See the figure below for further information:
Figure iv: The four quadrants of the unit circle
In quadrants I and IV, the X-coordinate (cosine) will always be positive, but negative in quadrants II and III. However, in quadrants I and II, the y-coordinate (sine) will always be positive, whereas in quadrants III and IV, it will always be negative.
- Learn How To Solve For Tangent: You must be excited about utilizing the unit circle to do it properly. You should concentrate on solving for tangent since it makes knowing the values for cosine and sine much easier. Below is the tangent formula: