Introduction to Circle
Let us know about the circle, before going through what is unit circle? or try to define a unit circle. A circle is the locus of all points which lie on a perimeter of equal distance from a given point. The point from which all the points on a circle are equidistant will be called the centre of the circle, and the distance from that centre to all the points on the circle will be called the radius of the circle. A diagram is shown below.
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The circle above has shown its centre at point C and a radius of length r. By definition, all radii of a circle are equal, since all the points on a circle are the same distance from the centre, and all the radii of a circle have one endpoint on the centre of the circle and one at the perimeter.
Chord and Its Properties
A chord of a circle is a straight line segment whose starting and end points both lie on the circle. A secant line is the infinite line extension of a chord and a chord is a line segment that attaches any two points on the curve, for example, an ellipse. The word chord is from the Latin ‘chorda’ and it implies bowstring.
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Chord Has Some Following Important Properties
Chords will be equidistant to the centre only if their lengths are equal.
Equal chords subtend equal angles to the centre of the circle.
A chord that passes through the centre of a circle will be called the diameter of the circle and it will be the longest chord.
Perpendicular drawn from the centre to the chord will divide the chord into two equal parts and vice versa.
What is Unit Circle?
Define Unit Circle - The unit circle is a circle of unit radius, that is a radius of 1. The unit circle in the trigonometry is the circle of radius 1 centred at the origin (0, 0) in the cartesian coordinate system of the euclidean plane.
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Consider if (x, y) is a point on the unit circles circumference, then |x| and |y| are the length of the legs of a right triangle, whose hypotenuse length is equal to 1. Thus, according to the Pythagorean theorem, a and b satisfies the equation
y2 + x2 = 1
Trigonometric Functions on the Unit Circle
The trigonometric functions like cos and sine of an angle are defined on the unit circle as given below.
If (x, y) is a point on the unit circle and a ray from the origin (0, 0) touches the point (x, y), then it makes an angle θ from the positive x-axis. In this clockwise turning is positive, then Cos θ = x and Sin θ = y.
x2 + y2 = 1 is the equation, which gives the relation
Cos2 θ + Sin2 θ = 1
The unit circle also shows that sine and cosine are periodic functions, with the identities
Cos θ = Cos(2π k + θ)
Sin θ = Sin(2π k + θ)
For any integer k.
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Diagram of the unit circle showing coordinate points.
Uses of Unit Circle
While working on the right triangles sine, cosine and other trigonometric functions are valid for angle measure more than zero or less than π/2. These functions produce meaningful values, when defined on the unit circles, for any real-valued angle measures. For example even those, greater than 2π. All the trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant, including archaic functions like versine and exsecant, can be defined geometrically in terms of a unit circle.
The unit circle is very useful, to calculate the values of any trigonometric functions, other than those labelled already, without using the calculator. Only by using the angle sum and difference formula, it can be calculated.
Facts About Circle
A circle has the shortest perimeter compared to all the shapes of the same area.
The shape of the circle has led to one of the very important invention wheels, which enabled our modern automotive system.
The shape of the circle looks attractive to humans, so it can be seen in many architectural structures.
FAQs on Unit Circle
1. What is a Circle?
Answer. A circle is a shape that consists of all the points in a single plane that are fixed at a constant distance from a given point, the centre. Equivalently we can say that it is the curve traced out by or the locus of a point that moves in a plane so that its distance from a given point is constant.
2. In How Many Parts a Circle Divides the Plane?
Answer. A circle divides any plane into three parts that are, the point inside the circle, the point outside the circle, these are also called the point exterior of the circle and the point on which the circle is drawn.
3. Mention Any Three Parts of a Circle.
Answer. The three important parts of the circle are radius, chord and diameter. Radius is the distance from the centre of the circle to its outer perimeter. A chord is a line segment whose endpoints lie on a circle. A diameter is a chord with a maximum length that passes through the centre of the circle.
