The unit circle is a foundational concept in mathematics and trigonometry, used to help calculate angles and measure lengths of lines. Understanding the basics of the unit circle is essential to increase your math skills, and we’ll be exploring what the unit circle is, breaking down the basics, and examining graphical and trigonometric interpretations. We’ll also review how to calculate angles in the unit circle and tips for remembering it.
What is the Unit Circle?
Understanding what the unit circle is will allow you to more effectively explore its properties. A unit circle is a two-dimensional circle with a radius of 1. The circle’s circumference is measured in radians, not degrees; one full rotation of the circle is equal to 2π (6.28) radians. It’s often represented by the equation
The Basics of the Unit Circle
The unit circle is divided into four quadrants, each representing the sine, cosine and tangent values for an angle of rotation. This is demonstrated in the coordinates of the unit circle. The bottom right quadrant represents 0 degrees (0π radians) , and each following clockwise quadrant increases by 90 degrees (π/2 radians). The top right quadrant is defined as 3π/2 or 270 degrees.The unit circle can also be understood geometrically; when a circle has a radius of 1, the trigonometric values of sine, cosine and tangent applied to it can all be found in a right triangle of length 1. It’s the same concept as Pythagoras’ Theorem, where you can use the sides of a triangle to calculate its hypotenuse.
Exploring the Unit Circle Graphically
Now that you understand the basics of the unit circle, let’s explore it graphically. The image on the left of this page shows a unit circle with lines dividing the Quadrants and points indicating angles at which each trigonometric function equals 0 or 1. Moving anti-clockwise from Quadrant I, you can see that sine equals 1, cosine equals 0, and tangent equals undefined (∞). In Quadrant II, sine equals 0, cosine equals -1 and tangent is undefined. In Quadrant III, sine equals -1, cosine equals 0 and tangent is undefined. And in Quadrant IV, sine equals 0, cosine equals 1, and tangent is undefined.
How to Interpret the Unit Circle Graphically
The graph has been created to help explain the trigonometric ratios of sine, cosine and tangent. Each point on the circumference of the circle corresponds to a specific angle. When these angles are measured in radians (not degrees), their sine, cosine, and tangent values can be calculated using geometry from the image above. These can then be graphed on a coordinate system and used for further analysis.
Analyzing the Unit Circle with Trigonometry
Once you understand how to interpret the unit circle graphically and what angles represent what points on the circle, it’s time to use this knowledge in trigonometry. Knowing the basics of trigonometry will help you find the exact values at any point on the unit circle, just by correlating specific angles with their corresponding trigonometric ratios. This can help when considering vectors, vectors component parts, distances from points, and more.
Using the Unit Circle to Calculate Angles
The unit circle is also useful for calculating angles in radians, rather than degrees. This is important for finding side lengths of triangles or other shapes and proving geometric proofs when necessary. Radians are measured by dividing the circumference of a circle by its radius and taking that number in degrees. Since we know that our unit circle has a radius of 1 at all points, we can use this to calculate the angle at each point in radians. To do this, we need to find out what angle corresponds with each x-axis component or y-axis component.
Understanding Radians and Degrees in the Unit Circle
Knowing how many degrees are equivalent to a given point on the unit circle helps us apply trigonometric ratios to that point. Both degrees and radians have their strengths; working in degrees gives a more concrete understanding of angles but radians are essential when working with circles or trigonometric functions (sine, cosine and tangent). To convert from degrees to radians, you need to know that 360° is equal to 2π. Hence, any angle measured in degrees can be converted into radians using this formula: θ = ° / 180 x π.
Applying the Unit Circle in Everyday Problems
It’s important to understand how to apply your knowledge on the unit circle to everyday problems. For example, if faced with an algebraic equation involving a triangle you can use your understanding of angles and trigonometry functions combined with calculations from the unit circle. You can use these methods to solve problems involving arc length or mean time conversions.
Tips for Remembering and Applying the Unit Circle
To apply the unit circle most effectively you need to have an accurate understanding of its graphical properties as well as its trigonometric properties. One way to further cement this knowledge is to remember some simple facts about it; for example, that sine is positive in Quadrants I & II, negative in Quadrants III & IV; that cosine is positive in Quadrant I & IV; that tangent is undefined in Quadrants I & III; and that sine is zero in Quadrants III & IV.
Learning about the unit circle can be challenging at first, but it doesn’t need to be overwhelming. With this article and its helpful images, you are now equipped with a better understanding of what a unit circle is and how you can use it for various problems in mathematics and trigonometry.
