A circle is a closed geometric figure with no sides. It consists of a center, and the distance from the center to the point lying on the circle is known as the radius. A circle can be drawn on a 2D plane. It represents the locus of points whose distance from a fixed point is a constant value. The equation of a circle can be represented in many forms, such as polar form, general form, standard form, and parametric form. The equation of a circle is used to find the position of the circle. The length of the circle and the coordinates of the center are basically required to form the equation of a circle. Let’s derive the equation of the circle. Let the coordinates of the center be (a, b), and (x, y) is an arbitrary point on the circumference of the circle.

Then using the Distance formula,

Now squaring both sides we get,

(x – a)² + (y – b)² = r²

Therefore the equation of circle is given by,

(x – a)² + (y – b)² = r²

Where (a, b) is the coordinates of the center and r is the radius. 

Unit Circle Formula

A unit circle is a circle of radius one. The coordinate of the center is (0, 0). The equation of the unit circle is given by,

(x – 0)² + (y – 0)²= 1

 

A unit circle has four quadrants. It is useful, especially in the field of trigonometry, as we can calculate the sine, cosine, and tangent functions. Suppose a right-angle triangle is drawn within the unit circle. Let the radius coordinate be (x, y). The x and y are lengths of base and altitude, respectively, and the hypotenuse is the radius.

Therefore sinθ = Altitude/Hypotenuse = y/1

cosθ = x/1

Therefore sin²θ + cos²θ = x² + y² = 1

A unit circle can also be used in the complex plane. Any complex number z = x + iy is said to lie on a unit circle with x² + y² = 1. Unit circle has a wide variety of applications. They are used to estimate heights and distances, used in trigonometry, engineering, spherical trigonometry, etc. 

Sample Problems

Question 1: Show that the point P(1/√3, 2/√3) lies on the unit circle

Solution:

Let x= 1/√3 y = 2/√3

Using the equation of circle we get x² = 1/3

y² = 2/3

x² + y²  = 1

Question 2: Prove cot² x +1 = cosec²  x using the unit circle

Solution:

As we know x² + y² =1 ⇢ (i)

cosec x = 1/sin x = 1/y

tan x = y/x

cot x = 1/tan x = x/y

Taking y² common from equation 1

y² (1 + x²/y²) = 1

=> (1 + x²/y²) = 1/y²

Hence proved

Question 3: Use the unit circle to find x and y when the angle of the right-angled triangle is 45 degrees and inscribed in the unit circle.

Solution:

For 45 degrees x and y are equal

x = y

x² + y² = 1

=> 2x² = 1

x= 1/ √2

Question 4: Prove (1, 0) lies on the unit circle.

Solution:

x² + y² = 1

1² + 0² = 1, which satisfies the equation of unit circle.

Question 5: Find x if y = 1/2 using the unit circle formula.

Solution:

The unit circle formula is x² + y² =1 

x² + 1/4 = 1

=> x = √3/2

Question 6: What point corresponds to the angle π/2 on the unit circle?

Solution:

x² + y² = 1 is the equation of the circle

y = sin θ

x = cos θ

cos ( π/2) = 0

sin (π/2) = 1

The point (0, 1) corresponds to 90°

Question 7: Find y if x = -1 using unit circle formula.

Solution:

The unit circle formula is x² + y² =1

y² + 1 = 1

=> y = 0