The Unit Circle

The unit circle is one of the most important concepts in trigonometry. It connects angles, coordinates, right triangles, and trigonometric functions in a single mathematical model.

A unit circle is a circle with a radius of exactly 1 unit, centred at the origin (0, 0) of a Cartesian coordinate system.

The unit circle allows us to determine exact values of sine, cosine, and tangent for common angles. It also helps explain why trigonometric functions can have positive and negative values and why their values repeat.

Unlike right-triangle trigonometry, which initially focuses on acute angles, the unit circle allows us to work with angles of any size, including angles greater than 90°, negative angles, and angles greater than 360°.

Defining the Unit Circle

The equation of a circle centred at the origin with radius r is:

x² + y² = r²

For a unit circle:

r = 1

Therefore:

x² + y² = 1

Every point (x, y) on the unit circle satisfies this equation.

For example, consider the point:

(0.6, 0.8)

Check:

0.6² + 0.8² = 0.36 + 0.64 = 1

Therefore, (0.6, 0.8) lies on the unit circle.

The point (2, 1) does not lie on the unit circle because:

2² + 1² = 5

which is not equal to 1.

Important Features

The unit circle has:

• Centre: (0, 0)

• Radius: 1 unit

• Diameter: 2 units

• Circumference: 2π units

• Equation: x² + y² = 1

The circle crosses the coordinate axes at:

(1, 0), (0, 1), (−1, 0), and (0, −1).

These four points correspond to the quadrantal angles 0°, 90°, 180°, and 270°.

Angles in Standard Position

An angle is in standard position when:

• Its vertex is at the origin.

• Its initial side lies along the positive x-axis.

• Its terminal side is determined by rotating from the initial side.

Angles are measured from the positive x-axis.

Positive angles are measured counterclockwise.

Negative angles are measured clockwise.

For example:

• 30° represents a counterclockwise rotation of 30°.

• 90° represents a quarter-turn counterclockwise.

• 180° represents a half-turn.

• 270° represents three-quarters of a full turn.

• −90° represents a quarter-turn clockwise.

A complete revolution corresponds to 360°.

Angles differing by a whole number of complete revolutions have the same terminal side.

These are called coterminal angles.

For example:

30°, 390°, and −330°

all have the same terminal side.

Degrees and Radians

Angles can be measured in degrees or radians.

A complete circle contains:

360° = 2π radians

Therefore:

180° = π radians

90° = π/2 radians

45° = π/4 radians

30° = π/6 radians

60° = π/3 radians

Converting Degrees to Radians

Use:

Radians = Degrees × π/180

Example:

Convert 120° to radians.

120 × π/180 = 2π/3

Therefore:

120° = 2π/3 radians

Converting Radians to Degrees

Use:

Degrees = Radians × 180/π

Example:

Convert 5π/6 to degrees.

(5π/6) × (180/π) = 150

Therefore:

5π/6 radians = 150°

Common Angle Conversions

Radians are especially important in calculus because the standard derivative and integral formulas for trigonometric functions assume angles are measured in radians.

Sine and Cosine on the Unit Circle

The most important relationship on the unit circle is:

(x, y) = (cos θ, sin θ)

where θ is the angle measured from the positive x-axis.

This means:

cos θ = x-coordinate

sin θ = y-coordinate

Every angle corresponds to a point on the unit circle.

Example: 30°

At an angle of 30°, the point on the unit circle is:

(√3/2, 1/2)

Therefore:

cos 30° = √3/2

sin 30° = 1/2

Example: 90°

At 90°, the point is:

(0, 1)

Therefore:

cos 90° = 0

sin 90° = 1

Example: 180°

At 180°, the point is:

(−1, 0)

Therefore:

cos 180° = −1

sin 180° = 0

The unit circle makes it possible to determine sine and cosine directly from coordinates.

Why the Coordinates Equal Cosine and Sine

Consider a point P(x, y) on the unit circle in the first quadrant.

Draw a perpendicular line from P to the x-axis.

This creates a right triangle with:

• Hypotenuse = 1

• Horizontal side = x

• Vertical side = y

Using right-triangle trigonometry:

cos θ = adjacent/hypotenuse

Since the hypotenuse is 1:

cos θ = x/1

cos θ = x

Similarly:

sin θ = opposite/hypotenuse

sin θ = y/1

sin θ = y

This relationship extends to all quadrants using signed coordinates.

It also explains why:

sin² θ + cos² θ = 1

This identity follows directly from the equation of the unit circle.

Exact Values for Special Angles

Certain angles have exact trigonometric values that can be found using special right triangles.

The most important are:

• 0°

• 30°

• 45°

• 60°

• 90°

The 45°–45°–90° Triangle

An isosceles right triangle has two equal legs.

If each leg has length 1, the hypotenuse is:

√(1² + 1²) = √2

To create a triangle with hypotenuse 1, divide each side by √2.

The legs become:

1/√2 = √2/2

Therefore:

sin 45° = √2/2

cos 45° = √2/2

The 30°–60°–90° Triangle

An equilateral triangle with side length 2 can be divided into two right triangles.

Each right triangle has:

• Hypotenuse = 2

• Shorter leg = 1

• Longer leg = √3

Dividing all sides by 2 gives a hypotenuse of 1.

Therefore:

sin 30° = 1/2

cos 30° = √3/2

sin 60° = √3/2

cos 60° = 1/2

Exact Trigonometric Values in the First Quadrant

These values are fundamental to trigonometry.

Rather than memorising every angle around the circle separately, we can use these first-quadrant values together with quadrant signs.

The Four Quadrants

The coordinate plane is divided into four quadrants.

Quadrant I

Angles between 0° and 90°.

x is positive.

y is positive.

Therefore:

• cos θ is positive.

• sin θ is positive.

• tan θ is positive.

Quadrant II

Angles between 90° and 180°.

x is negative.

y is positive.

Therefore:

• cos θ is negative.

• sin θ is positive.

• tan θ is negative.

Quadrant III

Angles between 180° and 270°.

x is negative.

y is negative.

Therefore:

• cos θ is negative.

• sin θ is negative.

• tan θ is positive.

Quadrant IV

Angles between 270° and 360°.

x is positive.

y is negative.

Therefore:

• cos θ is positive.

• sin θ is negative.

• tan θ is negative.

Quadrant Sign Summary

A useful memory aid is ASTC:

• All positive in Quadrant I.

• Sine positive in Quadrant II.

• Tangent positive in Quadrant III.

• Cosine positive in Quadrant IV.

The quadrant determines the sign of a trigonometric value, while the reference angle determines its magnitude.

Reference Angles

A reference angle is the acute angle between the terminal side of an angle and the x-axis.

Reference angles allow us to use familiar first-quadrant values to evaluate trigonometric functions in any quadrant.

For angles between 0° and 360°:

For quadrantal angles, the reference angle is 0°, and the axis coordinates should be used directly.

Worked Example: 150°

Find sin 150° and cos 150°.

150° lies in Quadrant II.

Reference angle:

α = 180° − 150° = 30°

In Quadrant II:

• Sine is positive.

• Cosine is negative.

Therefore:

sin 150° = 1/2

cos 150° = −√3/2

Worked Example: 225°

225° lies in Quadrant III.

Reference angle:

α = 225° − 180° = 45°

In Quadrant III, sine and cosine are both negative.

Therefore:

sin 225° = −√2/2

cos 225° = −√2/2

Worked Example: 300°

300° lies in Quadrant IV.

Reference angle:

α = 360° − 300° = 60°

In Quadrant IV:

• Cosine is positive.

• Sine is negative.

Therefore:

cos 300° = 1/2

sin 300° = −√3/2

Coordinates Around the Entire Unit Circle

The following table gives the exact coordinates for the most commonly used angles.

Remember:

(x, y) = (cos θ, sin θ)

The exact values follow a repeating pattern of magnitudes and signs.

Tangent on the Unit Circle

Tangent is defined by:

tan θ = sin θ / cos θ

Since:

sin θ = y

and:

cos θ = x

we can also write:

tan θ = y/x

provided x ≠ 0.

Worked Example: 60°

sin 60° = √3/2

cos 60° = 1/2

Therefore:

tan 60° = (√3/2)/(1/2)

tan 60° = √3

Worked Example: 135°

sin 135° = √2/2

cos 135° = −√2/2

Therefore:

tan 135° = (√2/2)/(−√2/2)

tan 135° = −1

When Is Tangent Undefined?

Tangent is undefined when cosine equals zero.

This occurs at:

90°, 270°, and coterminal angles.

For example:

tan 90° = 1/0

Division by zero is undefined.

Other Trigonometric Functions

The unit circle can also be used to evaluate cosecant, secant, and cotangent.

These functions are reciprocals of sine, cosine, and tangent.

csc θ = 1/sin θ

sec θ = 1/cos θ

cot θ = cos θ/sin θ

Worked Example: 30°

We know:

sin 30° = 1/2

cos 30° = √3/2

Therefore:

csc 30° = 1/(1/2)

csc 30° = 2

For secant:

sec 30° = 1/(√3/2)

= 2/√3

sec 30° = 2√3/3

For cotangent:

cot 30° = (√3/2)/(1/2)

cot 30° = √3

Undefined Values

• Cosecant is undefined when sine is zero.

• Secant is undefined when cosine is zero.

• Cotangent is undefined when sine is zero.

These restrictions are important when evaluating trigonometric expressions.

Negative Angles

Negative angles are measured clockwise from the positive x-axis.

For example:

−30° is coterminal with 330°.

Therefore:

cos(−30°) = cos 330° = √3/2

sin(−30°) = sin 330° = −1/2

This illustrates two important identities:

cos(−θ) = cos θ

sin(−θ) = −sin θ

Cosine is an even function.

Sine is an odd function.

Tangent is also odd:

tan(−θ) = −tan θ

Worked Example

Evaluate:

sin(−45°)

Since:

sin 45° = √2/2

Therefore:

sin(−45°) = −√2/2

Angles Greater Than 360°

Angles greater than 360° represent more than one complete revolution.

Because a complete revolution is 360°, we can subtract multiples of 360° to find a coterminal angle.

Worked Example: 420°

420° − 360° = 60°

Therefore:

sin 420° = sin 60°

sin 420° = √3/2

Worked Example: 765°

765° − 720° = 45°

Therefore:

cos 765° = cos 45°

cos 765° = √2/2

In radians, the corresponding period for sine and cosine is 2π.

Thus:

sin(θ + 2π) = sin θ

cos(θ + 2π) = cos θ

Tangent repeats every π radians, or 180°.

Trigonometric Identities from the Unit Circle

The unit circle provides a geometric explanation for several important trigonometric identities.

The Pythagorean Identity

The equation of the unit circle is:

x² + y² = 1

Since:

x = cos θ

y = sin θ

we obtain:

cos² θ + sin² θ = 1

This is the fundamental Pythagorean trigonometric identity.

Rearranging the Identity

We can write:

sin² θ = 1 − cos² θ

or:

cos² θ = 1 − sin² θ

These relationships allow us to calculate one trigonometric value when another is known.

Worked Example

Suppose:

cos θ = 3/5

and θ lies in Quadrant IV.

Find sin θ.

Use:

sin² θ + cos² θ = 1

Substitute:

sin² θ + 9/25 = 1

sin² θ = 16/25

Therefore:

sin θ = ±4/5

Since θ lies in Quadrant IV, sine is negative.

sin θ = −4/5

Using the Unit Circle to Solve Trigonometric Equations

The unit circle can help us find angles that satisfy trigonometric equations.

Worked Example: sin θ = 1/2

Find all solutions between 0° and 360°, including 0° but excluding 360°.

The reference angle is 30°.

Sine is positive in Quadrants I and II.

Therefore:

θ = 30° or 150°

Worked Example: cos θ = −√2/2

The reference angle is 45°.

Cosine is negative in Quadrants II and III.

Therefore:

θ = 135° or 225°

Worked Example: tan θ = −√3

The reference angle is 60°.

Tangent is negative in Quadrants II and IV.

Therefore:

θ = 120° or 300°

These methods are important when solving trigonometric equations in algebra, calculus, and physics.

Connecting the Unit Circle to Trigonometric Graphs

As an angle increases around the unit circle, the x-coordinate and y-coordinate change continuously.

These changes produce the cosine and sine graphs.

Sine Graph

The sine function represents the y-coordinate of the rotating point.

Its values range from −1 to 1.

Its period is:

360° or 2π radians

Cosine Graph

The cosine function represents the x-coordinate.

Its values also range from −1 to 1.

Its period is:

360° or 2π radians

Tangent Graph

Tangent is the ratio of sine to cosine.

Its values are not restricted to the interval from −1 to 1.

The tangent function has vertical asymptotes where cosine is zero.

Its period is:

180° or π radians

These relationships explain the repeating patterns of trigonometric functions.

Applications of the Unit Circle

The unit circle is used in many areas of mathematics, science, and engineering.

Circular Motion

A point moving around a circle of radius r can be described by:

x = r cos θ

y = r sin θ

For a unit circle, r = 1.

If the angular position changes at a constant rate ω:

θ = ωt + θ₀

then:

x(t) = r cos(ωt + θ₀)

y(t) = r sin(ωt + θ₀)

These equations describe uniform circular motion.

Waves and Oscillations

Sine and cosine functions describe many repeating phenomena, including:

• Sound waves

• Water waves

• Alternating electrical current

• Pendulum motion for small oscillations

• Vibrations in mechanical systems

A simple oscillation may be modelled by:

y(t) = A sin(ωt + φ)

where:

• A = amplitude

• ω = angular frequency

• φ = phase constant

• t = time

Engineering and Navigation

Sine and cosine can be used to resolve a vector into horizontal and vertical components.

For a vector with magnitude F and direction θ measured from the positive x-axis:

Fₓ = F cos θ

Fᵧ = F sin θ

These equations are used in force analysis, navigation, projectile motion, and engineering design.

Worked Example: Force Components

A force of 100 N acts at an angle of 30° above the positive horizontal axis.

Horizontal component:

Fₓ = 100 cos 30°

Fₓ = 100(√3/2)

Fₓ = 50√3 N ≈ 86.6 N

Vertical component:

Fᵧ = 100 sin 30°

Fᵧ = 100(1/2)

Fᵧ = 50 N

The unit circle provides the exact trigonometric values needed for the calculation.

A Reliable Strategy for Unit Circle Problems

When evaluating a trigonometric function:

1. Identify the angle.

2. Convert between degrees and radians if necessary.

3. Reduce the angle to a coterminal angle between 0° and 360°.

4. Determine the quadrant.

5. Find the reference angle.

6. Recall the exact value for the reference angle.

7. Apply the correct sign.

8. Simplify the result.

9. Check whether the function is defined.

Example

Evaluate:

cos 510°

Subtract 360°:

510° − 360° = 150°

150° lies in Quadrant II.

Reference angle:

180° − 150° = 30°

Cosine is negative in Quadrant II.

Therefore:

cos 510° = −√3/2

Common Mistakes

Confusing sine and cosine coordinates

On the unit circle, cosine is the x-coordinate and sine is the y-coordinate.

Forgetting that the radius is 1

The unit circle is specifically defined by a radius of one unit.

Mixing degrees and radians

Always check the angle measurement being used.

Using the wrong quadrant sign

Reference angles determine magnitude, but quadrants determine signs.

Assuming sine and cosine are always positive

Both functions can have negative values.

Confusing tangent with sine or cosine

Tangent is the ratio sin θ/cos θ.

Forgetting undefined values

Tangent and secant are undefined when cosine is zero. Cosecant and cotangent are undefined when sine is zero.

Using approximate decimals when exact values are required

For example, √3/2 is an exact value, while 0.866 is an approximation.

Forgetting coterminal angles

Angles greater than 360° or less than 0° can be reduced to familiar positions.

Assuming all trigonometric functions have the same period

Sine and cosine have period 360°, while tangent has period 180°.

Incorrectly applying the Pythagorean identity

The correct identity is:

sin² θ + cos² θ = 1

not sin θ + cos θ = 1.

Key Terms

Unit Circle: A circle of radius 1 centred at the origin.

Radius: The distance from the centre of a circle to any point on its circumference.

Origin: The point (0, 0) in the Cartesian coordinate plane.

Cartesian Coordinates: Ordered pairs (x, y) describing positions in a coordinate plane.

Angle in Standard Position: An angle whose vertex is at the origin and whose initial side lies along the positive x-axis.

Initial Side: The starting ray of an angle.

Terminal Side: The ray representing the final position after rotation.

Positive Angle: An angle measured counterclockwise.

Negative Angle: An angle measured clockwise.

Degree: A unit of angular measurement in which a complete revolution contains 360°.

Radian: A unit of angular measurement based on the ratio of arc length to radius.

Coterminal Angles: Angles that share the same terminal side.

Quadrant: One of the four regions of the coordinate plane.

Reference Angle: The acute angle between an angle's terminal side and the x-axis.

Sine: The trigonometric function corresponding to the y-coordinate on the unit circle.

Cosine: The trigonometric function corresponding to the x-coordinate on the unit circle.

Tangent: The ratio of sine to cosine.

Cosecant: The reciprocal of sine.

Secant: The reciprocal of cosine.

Cotangent: The reciprocal of tangent.

Exact Trigonometric Value: A trigonometric value expressed without rounding, often using fractions and radicals.

Pythagorean Identity: The relationship sin² θ + cos² θ = 1.

Trigonometric Identity: An equation involving trigonometric functions that is true wherever both sides are defined.

Periodic Function: A function whose values repeat at regular intervals.

Period: The length of one complete repeating cycle.

Amplitude: The maximum displacement from the midline of a periodic oscillation.

Angular Frequency: The rate of change of angular position, usually measured in radians per second.

Vector Component: The part of a vector acting along a specified coordinate direction.

Check Your Understanding

1. Define the unit circle and state its equation.

2. Determine whether the point (3/5, 4/5) lies on the unit circle.

3. Explain why the coordinates of a point on the unit circle are (cos θ, sin θ).

4. Convert 150° into radians.

5. Convert 7π/6 radians into degrees.

6. State the exact coordinates of the points corresponding to 0°, 90°, 180°, and 270°.

7. Find the exact values of sin 30°, cos 45°, and tan 60°.

8. Determine the exact coordinates of the point corresponding to 120°.

9. Find sin 225° and cos 225°.

10. Evaluate tan 150°.

11. Find sin(−60°) and cos(−60°).

12. Evaluate cos 750°.

13. If sin θ = 3/5 and θ lies in Quadrant II, determine cos θ.

14. Solve sin θ = −1/2 for 0° ≤ θ < 360°.

15. A force of 80 N acts at an angle of 60° above the horizontal. Use exact trigonometric values to determine its horizontal and vertical components.

Key Takeaways

• The unit circle is a circle of radius 1 centred at the origin.

• Its equation is x² + y² = 1.

• Every point on the unit circle can be represented as (cos θ, sin θ).

• Cosine gives the x-coordinate and sine gives the y-coordinate.

• Positive angles are measured counterclockwise.

• Negative angles are measured clockwise.

• A complete revolution is 360° or 2π radians.

• Coterminal angles share the same terminal side.

• Special angles such as 30°, 45°, and 60° have exact trigonometric values.

• Reference angles allow trigonometric values to be determined in all four quadrants.

• The quadrant determines whether sine, cosine, and tangent are positive or negative.

• Tangent is calculated using tan θ = sin θ/cos θ.

• Tangent is undefined when cosine equals zero.

• The Pythagorean identity sin² θ + cos² θ = 1 follows from the equation of the unit circle.

• Sine and cosine values are always between −1 and 1.

• Sine and cosine repeat every 360° or 2π radians.

• Tangent repeats every 180° or π radians.

• The unit circle provides a geometric explanation for trigonometric graphs and identities.

• The unit circle is essential for understanding circular motion, waves, oscillations, and vector components.

• Mastering the unit circle provides the foundation for advanced trigonometry, calculus, physics, and engineering mathematics.