Trigonometry becomes much easier when you understand the unit circle. The unit circle helps us define trigonometric ratios for all angles, including positive, negative, and angles greater than 90°.

What Is the Unit Circle?

A unit circle is a circle with:

• Centre at O(0,0)
• Radius equal to 1

The circle is drawn on the Cartesian plane with the x-axis and y-axis crossing at the centre. see the figure below

The Four Quadrants

The unit circle is divided into four sections called quadrants:

• First Quadrant (Quadrant I) → top right
• Second Quadrant (Quadrant II) → top left
• Third Quadrant (Quadrant III) → bottom left
• Fourth Quadrant (Quadrant IV) → bottom right

Positive and Negative Angles

Angles are measured from the positive x-axis.

• An angle measured anticlockwise is positive.
• An angle measured clockwise is negative.

Examples:

• 120° is a positive angle and lies in the second quadrant.
• -50° is a negative angle and lies in the fourth quadrant.

Determining Quadrants of Angles

To know where an angle lies:

First Quadrant

Angles between 0° and 90°

Example:
30° lies in Quadrant I

Second Quadrant

Angles between 90° and 180°

Example:
140° lies in Quadrant II

Third Quadrant

Angles between 180° and 270°

Example:
240° lies in Quadrant III

Fourth Quadrant

Angles between 270° and 360°

Example:
330° lies in Quadrant IV

Negative Angles

Negative angles move clockwise.

Example:
-70° lies in Quadrant IV
-120° lies in Quadrant III

The figure below shows angles of 120^o and -50^o marked on the unit circle. They are in the second and fourth quadrants respectively.

Determine which quadrants where 35^o, 45^o, 190^o, 280^o, 330^o,235^o are found.

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Coordinates on the Unit-Circle

One important idea about the unit circle is that every point on the circle represents:

(x, y) = (cos θ, sin θ)

This means:

• x-coordinate = cos θ
• y-coordinate = sin θ

Figure below is a unit-circle and angle PON=30°. Determine the values of x and y at point P.

Angle AON is a right-angled at N. Therefore:

A right-angled triangle is formed inside the circle.

Since the radius of the unit circle is 1:

OP = 1

Using trigonometric ratios:

Now for cosine:

Therefore, the coordinates of point P are:

P(0.86, 0.5)

Therefore, for a unit circle:

sinθ = y co-ordinates of P

cosθ = x co-ordinates of P.

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Key Ideas to Remember

• The unit circle has radius 1.
• Positive angles move anticlockwise.
• Negative angles move clockwise.
• Every point on the unit circle represents: (cos θ, sin θ)
• The x-coordinate gives cosine.
• The y-coordinate gives sine.

The unit-circle is the foundation for understanding trigonometric functions, graphing, and solving advanced trigonometry problems.

Related topics

• factorization of quadratic equation
• Introduction to quadratic equation and expression
• Unit circle