Unit Circle Assignment Help

Abstract

Learning trigonometry generally poses a relevant challenge for many high school students as it mainly impedes the access for the students regarding their careers in science, in mathematics, technology, and in engineering as this particular visuospatial model is generally called the unit circle which effectively acts like as an integrated conceptual based relevant structure that mainly supports for the solving problems that have encountered during the learning process, and gets transfer to the broader range of main problems in a particular domain. The related framework of the unit circle generally performed better than all those who did not report effectively using the better visualization of this.

Introduction to the Unit Circle

A unit circle has a radius of 1 as in other words, one can say that the distance from the center of the circle to any particular part of the edge would always be 1 as the unit doesn’t matter for this but inspect of this, one of the most important things about the unit circle is that it could effectively make several equations, and authentic calculations that are so simpler, and relevant. The unit circle also gets serve as a useful basis for looking at the related definitions of angles. A unit circle is simply get drawn around the origin (0,0) of the X, Y-axis, as a straight line which mainly drawn from the center point of the circle as along with the edge to the circle, the length of the line would be 1, and the diameter of the circle would 2 as because the diameter is twice of the radius, as the center point of the circle is mainly a point at where the x-axis and the y-axis intersect to each other, or either at the coordinates point of the circle.

This is mainly the triangulation concept which permits the mathematicians to extend up the sine, cosine, and tangent as through frequency which is outside of the right-angle triangle. The sine, cosine, and tangents are the ratios of the sides of the triangle to a given particular angle, which is commonly get referred to as theta.

Tangent: The ratio of the length of the opposite segment to the length of the adjacent segment.

Sine: This is the ratio between the lengths of the opposite side of the right triangle and the hypotenuse.

Cosine: It is the ratio of the length of the adjacent branch of a right triangle to the length of the hypotenuse.

Graphing Trigonometric Functions

Graphing the sine and cosine functions of trigonometry becomes much easier while the user is thinking of the unit circle because the x coordinate changes very easily as the user moves around the circle and starts starting with 1 and decreasing towards the value 1 to 180 degrees, then also increasing the same way, the sine function does the same thing, but it easily increases to its maximum value from 1 to 90 degrees, while following the same pattern. The tan graph requires efficient division of x by y, and thus it is more complex for the graph, and it also has key points where it is undefined.

Equation of a Unit Circle

The general equation of a circle is (x - a)² + (y - b)² = r², which generally represents that a circle is having its center (a, b), and also the radius r as this equation of a circle is effectively get simplified to represent the equation of the unit circle as this is because, a unit circle is effectively get formed with its center at the point (0, 0) that is the origin point of the coordinate axis named x, and y, and having a radius for 1 unit. This equation generally satisfied all the relevant points which generally get lying on the circle as across four main quadrants.

Finding the trigonometric functions as through using a unit circle

The user can calculate the trigonometric functions of sine, cosine, and tangent through using a unit circle, and for better understanding the trigonometric functions, the user could easily get imply the Pythagoras theorem in a unit circle, and for this, the user simply needs to consider a right triangle placed in the unit circle in the coordinate plane of cartesian as the radius of the circle effectively represents the hypotenuse of the right triangle, and the radius vector generally makes an angle theta with the positive x-axis, and the coordinates of the endpoint of the radius vector are (x, y) as at where, the values of x and y are the lengths of the base, and an altitude of the right triangle so, as now, the user have the right triangles with the sides 1, x, and y so, in this, the user could effectively apply the trigonometry, so, that with the help of this, they could easily find the values of trigonometric ratios.

cos θ = x and sin θ =y as now the user has this value, which he could easily obtain from the values of other trigonometric ratios through using the right-angled triangle as within the unit circle, and if the user wants for obtaining the principal values of all these trigonometric ratios, then they can easily obtain this as through changing the values of theta of the related values.

Unit circle chart in radians

The unit circle generally represents a complete angle for 2π radians, and this unit circle is divided into four quadrants, and in the first quadrant at some angles, there is an availability of the standard values which are effectively getting applicable with the trigonometric ratios. The main points on the unit circle for the availability of all these angles generally represent the standard angle values of the sine and cosine values. The close observation of all these figures are the main values are generally repeated across the entire four quadrants but also with the change in the signs, as this change in sign is mainly due to the reference x-axis, and y-axis which are positive on one side, and negative from the other side of the origin. With the help of this, the user could easily find the relevant, and authentic trigonometric ratio values for the standard angles, across the four quadrants of the unit circle.