Unit Circle Formula

Before learning the unit circle formula, let us recall what is a unit circle. The unit circle is a circle centered at the origin, (0, 0) and its radius is 1. The unit circle formula helps to find the missing coordinates on the unit circle. The unit circle plays an important role in trigonometry. Let us study the unit circle formula with solved examples in the following sections.

What Is Unit Circle Formula?

In general, the equation of a circle whose center is \((x_1, y_1)\) and whose radius is r is given by the formula \(\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}=r^{2}\). As we have learned earlier, the center of the unit circle is \((x_1, y_1)\) = (0, 0). The radius of the unit circle is, r = 1 unit. Substituting these values in the above formula, the equation of the unit circle is:

(x - 0)² + (y - 0)² = 1²

x² + y² = 1

Therefore, the unit circle formula is:

x² + y² = 1

where,

• (x, y) are the coordinates of any point lying on the unit circle.

Let us see the applications of the unit circle formula in the following section.

Solved Examples Using Unit Circle Formula

Example 1: Does the point P (1/2, 1/2) lie on the unit circle?

Solution:

To find: whether point P lies on the unit circle.

The given point is: P (1/2, 1/2) = (x, y).

Let us assume that the point lies on the unit circle, which means the coordinates of the given point must satisfy the unit circle formula condition.

Substitute x = 1/2 and y = 1/2 in the unit circle formula,

x² + y² = 1

(1/2)² + (1/2)² = 1

1/4 + 1/4 = 1

1/2 = 1, which is false

Hence, our assumption that the point lies on the unit circle is wrong.

Answer: The point P does not lie on the unit circle.

Example 2: If P (√3/2, y) lies on the unit circle in the fourth quadrant, find y.

Solution:

To find: value of y.

The given point is, P (√3/2, y) = (x, y).

Substitute x = √3/2 in the unit circle formula,

x² + y² = 1

(√3/2)² + y² = 1

3/4 + y² = 1

y² = 1/4

y = ± 1/2

It is given that P lies in the fourth quadrant where the y-coordinate is negative.

Hence, y = - 1/2

Answer: y = -1/2.