Unit Circle

The unit circle is an important and key concept of trigonometry. This will introduce too many connections with the various and significant parts of trigonometry. This concept is a super important topic. The unit circle is used so often. Concept of the unit circle is so simple and easy. now the question is what is the unit circle?  so answer for this question is “it is a circle with radius or which have a radius equal to 1 unit”. The unit circle gives the right angle triangles which have the same hypotenuse equals to one always. it also helps us in deriving sin2 θ + cos2 θ = 1. If we draw now to make it easier to understand we will draw a horizontal line and named it as x-axis, and then we will draw a verticle line and named it as y-axis, in simple on plain paper draw x-axis and y-axis it has the centre which we called origin where x-axis and y-axis meet. Now draw a circle, as we know the radius is 1 unit and mark the point where both the axes are meeting we called it origin. Mark the origin with ‘p’. the circle which we draw meet with the x-axis at two points and y-axis at two points. Below We have provided a figure from which you can easily learn the written format.

unit circle

We must also be aware of radians and coordinates what are these? so in very easy language, the radians helps us in measuring angles and it is explained by the arc of a circle. Radius is equal to the arc of the circle, a full circle is of 2π  radians. the coordinate is the line from the origin to the circumference of the circle touching the axes. As we discussed above our circle is meeting with both the axes at two points. now we will find the coordinates with the help of these points.

How to find the Coordinates Of A Unit Circle?

  1. On the x-axis we will mark another point on the circumference of the circle drawn so from the origin which is ‘p’ to the marked point on the circumference we will get the x coordinate.
  2.  The radius of the circle is one then our x- coordinate will be one. the coordinates of this point will be written as (1, 0) and by moving towards the left side of the x-axis which is negative we will have (-1, 0).
  3. Now moving ahead on the y-axis from the origin ‘p’ to the y-axis the point marked on the circumference of the circle the coordinate will be written as (0, 1) keep in mind the radius is one.
  4. And on the same axis, we will move downward the negative side of y-axis we will get the coordinates like (0, -1) this is called the Unit circle. same as the figure given down below.
unit circle solution

This unit circle concept also helps us to recognize the other trigonometric features it also includes  ‘θ‘ we have three θ’s, Sinθ, Cosθ and Tanθ. Now again we draw x-axis and y-axis and the circle, we will draw a right triangle PQR. using origin ‘p’ and x-axis, a triangle PQR has hypotenuse, adjacent and opposite. PQ is our hypotenuse, PR will be the adjacent, QR will be the opposite. Let’s take the hypotenuse is equal to the radius of the triangle that means it is equal to one,  and now For the Sin theta the formula we use is Sinθ = opposite/hypotenuse = QR/QP, so we will get QR/1 because QP is hypotenuse which is equal to radius one. The result for Sin theta is QR. Now comes to the Cosθ = adjacent/hypotenuse = PR/QP, here also hypotenuse is equal to one. The result for Cosθ will be PR. Now the question arises for tanθ what tanθ will be? So for tanθ, it is defined as a tangent function with the help of a diagram it becomes quite easy to understand tanθ from the point Q draw a line towards the x-axis and that line is tangent. Mark the Sinθ, Cosθ and Tanθ on the triangle what we see our QR the opposite line, becomes Sinθ, PR, adjacent becomes Cosθ, and from point Q the line towards the x-axis is Tanθ.  It covers various conditions first one is if we increase θ to the upward direction near y-axis our sin and tan is increasing while cos is decreasing and in the second condition we will decrease the θ downwards towards the x-axis we will see the cos is increasing and tan is decreasing. In both the conditions the radius will remain the same it is not changing. Now in the third condition, if we increase the length of θ to the 90 degrees what will happen to sin and cosθ, cosθbecomes zero and sinθ becomes 1 the radius of the circle, and we can not find tanθ in this condition.hence the unit circle helps in knowing various points of Pythagoras theorem also. This will be proved to be helpful and easy to understands. You might also love mathway solver for your other mathematics problems. Hope you will love the procedure and we wish a very tapping unit circling to you.