Trigonometric Functions Using Unit Circle
The application of the Pythagoras theorem in a unit circle can be better used to understand trigonometric functions. For this, we consider a right triangle to be placed inside a unit circle in the Cartesian coordinate plane. If we notice, the radius of this circle denotes the hypotenuse of the right-angled triangle.
The radius of the circle forms a vector. This leads to the formation of an angle, say θ with the positive x-axis. Let us suppose x to be the base length and y to be the altitude length of the right triangle respectively. Also, the coordinates of the radius vector endpoints are (x, y) respectively.
The right-angle triangle holds the sides 1, x, and y respectively. The trigonometric ratio can be computed now, as follows:
sin θ = Altitude/Hypotenuse = y/1
cos θ = Base/Hypotenuse = x/1
Now,
- sin θ = y
- cos θ = x
- tan θ = sin θ /cos θ = y/x
On substituting the values of θ, we can obtain principal values of all the trigonometric functions. Simillarly values of trigonometric functions at different values is found.
Unit Circle: Definition, Formula, Diagram and Solved Examples
Unit Circle is a Circle whose radius is 1. The center of unit circle is at origin(0,0) on the axis. The circumference of Unit Circle is 2π units, whereas area of Unit Circle is π units2. It carries all the properties of Circle. Unit Circle has the equation x2 + y2 = 1. This Unit Circle helps in defining various Trigonometric concepts.
The Unit Circle is often denoted as S1 generalization to higher dimensions is the unit sphere. Let’s understand more about Unit Circle, Formula and Solved examples in detail below.