The Unit Circle

Master the unit circle and find trigonometric values. Grade 12 Math 30-1.

Lesson 4.2 of Trigonometric Ratios & Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Unit Circle & Its Equation. Discover the Unit Circle with radius 1, understand the equation x² + y² = 1, and master the four key anchor points that form its foundation.
• Building the Unit Circle Using Special Triangles. Use 45°–45°–90° and 30°–60°–90° triangles to find exact coordinates on the Unit Circle for 30°, 45°, and 60° angles.
• The Complete Unit Circle. See all the common angles on the Unit Circle with their degree measures, radian measures, and exact coordinates.

Inside the lesson — a free preview

Definition

The Unit Circle is a circle:

• With radius r = 1

• Centered at the origin (0, 0)

• Drawn on the Cartesian plane

Why Is It Called the "Unit" Circle?

Because its radius is exactly:

r = 1

Every point on the circle is exactly 1 unit away from the origin.

Unit Circle Equation

What You're Looking At. The diagram shows:

• A circle with radius = 1 centered at the origin O

• A point P(x, y) somewhere on the circle

• A right triangle formed by dropping a vertical line from P down to point A on the x-axis

Breaking Down the Triangle. The right triangle has three sides:

1. Horizontal leg (O to A): This is the x-coordinate of point P. It shows how far left or right P is from the y-axis.

2. Vertical leg (A to P): This is the y-coordinate of point P. It shows how far up or down P is from the x-axis.

3. Hypotenuse (O to P): This is the radius of the circle, which is always 1 on the Unit Circle.

The Key Insight. Since we have a right triangle, we can use the Pythagorean Theorem:

(horizontal leg)^2 + (vertical leg)^2 = (hypotenuse)^2

Substituting our values:

x^2 + y^2 = 1^2

The Unit Circle Equation. Simplifying:

x^2 + y^2 = 1

This is the equation of the Unit Circle!

No matter which point P(x, y) you pick on the circle, its coordinates will always satisfy this equation because the distance from the origin is always 1.

Worked example: Finding Coordinates from Angles

If Q is the point on the terminal arm of angle α in standard position on the unit circle, find the exact coordinates of Q for each of the following angles.

a) α = (π)/(3)

Solution:

Find (π)/(3) on the unit circle diagram above. Looking at the circle, you can see that the coordinates are:

Q = ((1)/(2), (√(3))/(2))

b) α = (5π)/(6)

Solution:

Find (5π)/(6) on the unit circle diagram above. This angle is in the second quadrant, and you can see that the coordinates are:

Q = (-(√(3))/(2), (1)/(2))

Notice that in the second quadrant, the x-coordinate is negative (to the left of the y-axis) while the y-coordinate is positive (above the x-axis).

c) α = -(π)/(4)

Solution:

A negative angle means we rotate clockwise from the positive x-axis. Rotating -(π)/(4) clockwise is the same as (7π)/(4) counterclockwise. Find (7π)/(4) on the unit circle diagram above (in the fourth quadrant), and you can see:

Q = ((√(2))/(2), -(√(2))/(2))

d) α = (7π)/(4)

Solution:

Find (7π)/(4) on the unit circle diagram above. This is the same angle as part (c)! Looking at the circle in the fourth quadrant, you can see:

Q = ((√(2))/(2), -(√(2))/(2))

This demonstrates that different angle measures can correspond to the same point on the unit circle.

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