Sine and cosine as coordinates on the unit circle — extending trigonometry from triangle ratios into a full function system that describes rotation, waves, and periodic motion
Chapter 3 introduced angles as rotations, radian measure as a natural way to quantify those rotations, and arc length as a connection between angle and circle. Now it is time to ask a deeper question: what happens at the point where a rotating ray meets a circle?
Chapter 4 builds the unit circle as the foundation for trigonometric functions. The unit circle is not merely a diagram to memorize. It is the definition of sine and cosine as functions of rotation. Once you understand the unit circle, trigonometry transforms from a collection of triangle ratios into a unified function system that describes periodic motion, waves, rotation, oscillation, and circular behavior throughout mathematics, science, and engineering.
This chapter should be read carefully and slowly. The unit circle is one of the most important structures in all of precalculus. Every section of Chapter 5 (trigonometric graphs), Chapter 6 (identities and equations), and many sections of Chapters 7, 12, and 13 depend on understanding this chapter well.
A point starts at (1, 0) on a circle of radius 1 and travels counterclockwise. After rotating through an angle of π/3 radians, where is the point? What are its coordinates? How do those coordinates relate to the sine and cosine of the angle? By the end of this chapter, you will answer this question precisely and extend it to every angle on the full unit circle.
In Algebra II and in right-triangle trigonometry, the sine, cosine, and tangent of an angle were defined as ratios of side lengths in a right triangle. That definition works, but it has serious limitations. Right triangles have acute angles only — angles between 0° and 90° can be described by triangle ratios, but what about 135°? What about 270°? What about a negative angle or one greater than 2π? Right triangles cannot describe those cases.
Trigonometry needs a definition that works for every angle — positive, negative, acute, obtuse, reflex, and beyond. The unit circle provides that definition. On the unit circle, sine and cosine are coordinates, and this definition extends seamlessly to every real-number angle.
In earlier mathematics, trigonometry was introduced through right triangles. Given a right triangle with an acute angle θ, the three primary trigonometric ratios were defined as:
sin(θ) = opposite / hypotenuse cos(θ) = adjacent / hypotenuse tan(θ) = opposite / adjacent
These definitions are useful for solving triangles and computing distances, and they will return in Chapter 7. However, they have a fundamental limitation: they only work for angles strictly between 0° and 90°. A right triangle cannot have an angle of 120°, 270°, or −π/4.
For trigonometry to become a true function system — one that accepts any real number as input and produces an output — it must be freed from the right triangle. The unit circle achieves this.
The choice of radius 1 is deliberate. On any circle of radius r, the sine of an angle could be defined as y/r and the cosine as x/r, where (x, y) is the point on the circle. When r = 1, those fractions simplify to just y and x directly. The unit circle makes the connection between trigonometric functions and coordinates as clean as possible.
When sine and cosine are defined using the unit circle, they become genuine functions of a real-number input. The input is an angle (in radians or degrees), and the output is a coordinate value between −1 and 1. This unlocks a rich function theory:
| Property | Sine and Cosine |
|---|---|
| Domain | All real numbers: (−∞, ∞) |
| Range | [−1, 1] |
| Graph shape | Smooth, continuous waves |
| Period | 2π (repeats every full rotation) |
The unit circle is the circle centered at the origin with radius 1. Its equation is:
x² + y² = 1
Every point (x, y) on the unit circle satisfies this equation, and every point satisfying this equation lies on the unit circle.
Place an angle θ in standard position: its initial side along the positive x-axis, rotating counterclockwise for positive θ and clockwise for negative θ. The terminal side of this angle intersects the unit circle at exactly one point. That intersection point is defined to have coordinates (cos θ, sin θ).
This is the core definition. Read it carefully:
For any angle θ in standard position, the terminal side intersects the unit circle at the point (x, y) where:
x = cos θ (the horizontal coordinate)
y = sin θ (the vertical coordinate)
Draw the unit circle centered at the origin. Draw a ray from the origin at angle θ measured counterclockwise from the positive x-axis. Mark the intersection point (x, y) on the circle. Label x = cos θ and y = sin θ. Draw perpendicular dashed lines from the point to both axes to show the coordinate values. Label the radius = 1.
This definition is consistent with the right-triangle definition for angles between 0° and 90°. When θ is acute, the terminal side and the x-axis form a right triangle whose hypotenuse is the unit circle radius = 1. In that triangle, sin θ = opposite/1 = y and cos θ = adjacent/1 = x. The unit circle definition reduces to the right-triangle definition in the acute case and extends naturally beyond it.
Since every point on the unit circle satisfies x² + y² = 1, and the coordinates are cos θ and sin θ, substituting gives the most important identity in trigonometry:
cos²θ + sin²θ = 1
This is the Pythagorean identity, and it holds for every angle θ without exception. It comes directly from the equation of the unit circle.
Before filling in special angles, it helps to identify the four points where the unit circle crosses the coordinate axes.
| Angle (degrees) | Angle (radians) | Point (x, y) | Notes |
|---|---|---|---|
| 0° | 0 | (1, 0) | Positive x-axis |
| 90° | π/2 | (0, 1) | Positive y-axis |
| 180° | π | (−1, 0) | Negative x-axis |
| 270° | 3π/2 | (0, −1) | Negative y-axis |
| 360° | 2π | (1, 0) | Full rotation; same as 0° |
Because every point on the unit circle satisfies x² + y² = 1, both coordinates must satisfy −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1. This means the values of sine and cosine are always between −1 and 1 inclusive. They never exceed 1 or fall below −1.
Students sometimes write sin θ = 2 or cos θ = −3 when solving equations. These are impossible values. If your work produces a sine or cosine outside [−1, 1], something has gone wrong. No angle on the unit circle produces such a result.
For any angle θ in standard position, let (x, y) be the point where the terminal side of θ meets the unit circle. Then:
cos θ = x (the horizontal coordinate)
sin θ = y (the vertical coordinate)
Domain: all real numbers (−∞, ∞) Range: [−1, 1]
Think of a point traveling counterclockwise around the unit circle. As the point moves, its x-coordinate oscillates between −1 and 1 — this oscillation is the cosine function. Its y-coordinate oscillates between −1 and 1 as well — this is the sine function. The two coordinates trace out the same oscillating pattern but offset from each other: cosine starts at its maximum (x = 1 when θ = 0), while sine starts at 0 (y = 0 when θ = 0). This offset relationship becomes central in Chapter 5.
cos θ = −√2/2 sin θ = √2/2
This corresponds to θ = 135° (or 3π/4)
The tangent of angle θ is the ratio of sine to cosine:
tan θ = sin θ / cos θ = y / x (provided x ≠ 0)
Domain: all real numbers except θ = π/2 + nπ for any integer n Range: (−∞, ∞)
Notice that y/x is the slope of the line from the origin to the point (x, y). The terminal side of θ passes through the origin and the unit circle point (cos θ, sin θ). Its slope is:
slope = (sin θ − 0) / (cos θ − 0) = sin θ / cos θ = tan θ
So tan θ is the slope of the terminal side of θ. This explains why tangent is undefined at θ = π/2 and θ = 3π/2 — the terminal sides there are vertical lines, which have undefined slope.
| Function | Definition | In Terms of (x, y) | Undefined When | Range |
|---|---|---|---|---|
| Secant | 1 / cos θ | 1 / x | cos θ = 0 (θ = π/2 + nπ) | (−∞,−1] ∪ [1,∞) |
| Cosecant | 1 / sin θ | 1 / y | sin θ = 0 (θ = nπ) | (−∞,−1] ∪ [1,∞) |
| Cotangent | cos θ / sin θ | x / y | sin θ = 0 (θ = nπ) | (−∞, ∞) |
The notation sin⁻¹(x) does not mean 1/sin(x). It means the inverse sine function (arcsine), which gives the angle whose sine is x. The reciprocal of sine is written csc(x). These are completely different functions. Never confuse them.
sec θ = 1 / cos θ = 1 / (1/2) = 2
csc θ = 1 / sin θ = 1 / (√3/2) = 2/√3 = 2√3/3
cot θ = cos θ / sin θ = (1/2) / (√3/2) = 1/√3 = √3/3
The unit circle is divided into four quadrants by the coordinate axes. The signs of the trigonometric functions depend entirely on which quadrant the terminal side of θ falls in.
| Quadrant | x (cos) | y (sin) | tan | Positive Functions |
|---|---|---|---|---|
| I (0 to π/2) | + | + | + | All six |
| II (π/2 to π) | − | + | − | sin, csc |
| III (π to 3π/2) | − | − | + | tan, cot |
| IV (3π/2 to 2π) | + | − | − | cos, sec |
Going counterclockwise from Quadrant I: All, Sine, Tangent, Cosine. These are the primary functions that are positive in each quadrant. Some students remember this as "All Students Take Calculus."
Draw the unit circle. Divide it into four quadrants with labeled axes. In each quadrant, write the sign pattern for sin and cos (e.g., "sin +, cos −" in Quadrant II) and the ASTC label indicating which function is positive. Add representative points in each quadrant showing positive and negative coordinate values.
(a) 200° is between 180° and 270° → Quadrant III → sin is NEGATIVE
(b) 5π/4 = 225° → Quadrant III → cos is NEGATIVE
(c) 315° is between 270° and 360° → Quadrant IV → tan is NEGATIVE
(d) 2π/3 = 120° → Quadrant II → cos is negative → sec = 1/cos is also NEGATIVE
For each angle, name the quadrant and state whether sin, cos, and tan are positive or negative.
Most angles on the unit circle do not have coordinates expressible as simple fractions or square roots. However, certain angles have unit circle coordinates that can be computed exactly using geometry. These are called special angles, and they correspond to the 30-60-90 and 45-45-90 triangle structures.
A right triangle with two 45° angles and hypotenuse 1 has both legs equal to √2/2 (by the Pythagorean theorem). Placing this triangle inside the unit circle with the angle at the origin, the far vertex lands at (√2/2, √2/2). Therefore for θ = 45°: cos(45°) = √2/2 and sin(45°) = √2/2.
A right triangle with angles 30°-60°-90° and hypotenuse 1 has the shorter leg opposite 30° equal to 1/2, and the longer leg opposite 60° equal to √3/2. Placing each angle at the origin gives the coordinates below.
| Angle | Radians | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
Reading sine from 0° to 90°, the values are √0/2, √1/2, √2/2, √3/2, √4/2 — which simplify to 0, 1/2, √2/2, √3/2, 1. The cosine values follow the same pattern in reverse. Understanding the geometric source (30-60-90 and 45-45-90 triangles) is more important than memorization, but this pattern can help as a check.
In calculus, physics, and most advanced applications, radians are the preferred unit. Every special angle has an exact radian equivalent. The five Quadrant I reference angles in radians are 0, π/6, π/4, π/3, π/2 — corresponding to 0°, 30°, 45°, 60°, 90°. Once the Quadrant I values are known, all four quadrants follow from the reference angle method and sign patterns.
π + π/3 = 3π/3 + π/3 = 4π/3The reference angle is the acute angle between the terminal side of θ and the nearest part of the x-axis. It is always between 0 and π/2 and always positive. Reference angles unlock the entire unit circle from just the Quadrant I values.
Step 1. Find the reference angle θ̂ for the given angle θ.
Step 2. Evaluate the trig function at θ̂ using Quadrant I values.
Step 3. Determine the sign based on which quadrant the original angle θ is in.
Step 4. Combine the magnitude from Step 2 with the sign from Step 3.
| Quadrant | Range of θ | Reference Angle θ̂ |
|---|---|---|
| I | 0 < θ < π/2 | θ̂ = θ |
| II | π/2 < θ < π | θ̂ = π − θ |
| III | π < θ < 3π/2 | θ̂ = θ − π |
| IV | 3π/2 < θ < 2π | θ̂ = 2π − θ |
Draw four separate unit circle diagrams, one for each quadrant. In each, show the terminal side of an angle θ, the reference angle θ̂ between the terminal side and the nearest x-axis, and the reference triangle formed by dropping a perpendicular to the x-axis. Label the reference angle in each diagram.
210° is in Quadrant III (between 180° and 270°)
Reference angle: 210° − 180° = 30°
|sin(210°)| = sin(30°) = 1/2
In Quadrant III, sine is NEGATIVE → sin(210°) = −1/2
7π/4 = 315° → Quadrant IV
Reference angle: 2π − 7π/4 = π/4
|cos(7π/4)| = cos(π/4) = √2/2
In Quadrant IV, cosine is POSITIVE → cos(7π/4) = √2/2
Find the reference angle and then evaluate the given function exactly.
Step 1. Identify the quadrant (or axis) of the terminal side of θ.
Step 2. Find the reference angle θ̂.
Step 3. Find exact coordinates (cos θ̂, sin θ̂) from Quadrant I values.
Step 4. Apply the correct signs for the quadrant to get (cos θ, sin θ).
Step 5. Compute tangent and reciprocal functions from the coordinates.
| Angle | Point | sin | cos | tan |
|---|---|---|---|---|
0 | (1, 0) | 0 | 1 | 0 |
π/2 | (0, 1) | 1 | 0 | undefined |
π | (−1, 0) | 0 | −1 | 0 |
3π/2 | (0, −1) | −1 | 0 | undefined |
cos(5π/6) = −√3/2 sin(5π/6) = 1/2
tan(5π/6) = sin/cos = (1/2)/(−√3/2) = −1/√3 = −√3/3
sec(5π/6) = 1/cos = 1/(−√3/2) = −2/√3 = −2√3/3
csc(5π/6) = 1/sin = 1/(1/2) = 2
cot(5π/6) = cos/sin = (−√3/2)/(1/2) = −√3
A calculator can approximate sin(30°) ≈ 0.5000000. But exact values carry more precision and meaning than approximations. Working with exact values — fractions, radicals, and π — is essential in algebra, calculus, and physics, where approximate decimals compound errors and obscure structure. When an answer is √3/2, leaving it in that form is correct and complete.
The unit circle has 16 standard positions defined by the special angles in each quadrant. Their exact coordinates are given below, organized by quadrant.
| Quadrant | Angle (radians) | Angle (degrees) | cos θ | sin θ |
|---|---|---|---|---|
| I & Axes | 0 | 0° | 1 | 0 |
π/6 | 30° | √3/2 | 1/2 | |
π/4 | 45° | √2/2 | √2/2 | |
π/3 | 60° | 1/2 | √3/2 | |
π/2 | 90° | 0 | 1 | |
| II | 2π/3 | 120° | −1/2 | √3/2 |
3π/4 | 135° | −√2/2 | √2/2 | |
5π/6 | 150° | −√3/2 | 1/2 | |
π | 180° | −1 | 0 | |
| III | 7π/6 | 210° | −√3/2 | −1/2 |
5π/4 | 225° | −√2/2 | −√2/2 | |
4π/3 | 240° | −1/2 | −√3/2 | |
3π/2 | 270° | 0 | −1 | |
| IV | 5π/3 | 300° | 1/2 | −√3/2 |
7π/4 | 315° | √2/2 | −√2/2 | |
11π/6 | 330° | √3/2 | −1/2 | |
2π | 360° | 1 | 0 |
Draw the full unit circle. Mark all 16 standard positions with dots. At each position, label the radian angle measure (outside the circle) and the exact (x, y) coordinates. Use a clean layout with labeled axes, quadrant numbers, and the circle drawn to scale. This is the complete unit circle diagram students should be able to reproduce.
tan(4π/3) = sin(4π/3) / cos(4π/3) = (−√3/2) / (−1/2) = (−√3/2) · (−2/1) = √3
Evaluate each trigonometric function exactly. Do not use a calculator.
Answers may be placed in the answer key.
If (x, y) is the point on the unit circle at angle θ, then the reflection of this point across the x-axis is (x, −y), which is the point at angle −θ. Therefore:
cos(−θ) = x = cos(θ) → cosine is an EVEN function: f(−x) = f(x)
sin(−θ) = −y = −sin(θ) → sine is an ODD function: f(−x) = −f(x)
tan(−θ) = sin(−θ)/cos(−θ) = −sin(θ)/cos(θ) = −tan(θ) → tangent is ODD
An even function has a graph symmetric across the y-axis. An odd function has a graph symmetric about the origin. Cosine is even; sine and tangent are odd. Secant is even; cosecant and cotangent are odd.
If (x, y) is at angle θ, its reflection across the y-axis is (−x, y), which is at angle π − θ. This gives:
cos(π − θ) = −cos(θ) sin(π − θ) = sin(θ)
This explains why sin(150°) = sin(30°) = 1/2: the points 30° and 150° are reflections across the y-axis, sharing the same y-coordinate.
cos(θ + π) = −cos(θ) sin(θ + π) = −sin(θ)
This explains why sin(210°) = −sin(30°): the points 30° and 210° are reflections through the origin, with both coordinates negated.
The unit circle has a complete cycle every 2π radians. After rotating 2π, the terminal side returns to the same position and the same unit circle point.
cos(θ + 2π) = cos(θ) Cosine has period 2π.
sin(θ + 2π) = sin(θ) Sine has period 2π.
tan(θ + π) = tan(θ) Tangent has period π.
sin(−π/3) = −sin(π/3) = −√3/2
cos(7π/3) = cos(2π + π/3) = cos(π/3) = 1/2
Use symmetry properties or periodicity to evaluate each expression without a reference table.
Correction: cos θ = x (horizontal) and sin θ = y (vertical). Cosine is the x-coordinate; sine is the y-coordinate. Cosine reaches its maximum of 1 at θ = 0, where the point is at (1, 0) on the positive x-axis.
Correction: All points on the unit circle satisfy x² + y² = 1, so x and y are always in [−1, 1]. No angle produces a sine or cosine outside this interval. If an equation leads to sin θ = 2, the equation has no solution.
Correction: 240° is in Quadrant III where sine is negative. The reference angle is 60°, so |sin(240°)| = sin(60°) = √3/2, but with the Quadrant III sign: sin(240°) = −√3/2. Always check the sign after computing the magnitude.
Correction: sin⁻¹(x) is the inverse sine function (arcsine), giving the angle whose sine is x. The reciprocal of sine is csc(x). The superscript −1 on a trigonometric function name always means inverse function, not reciprocal.
Correction: In Quadrant III (π < θ < 3π/2), the reference angle is θ − π (or θ − 180°), which is positive because θ > π. The reference angle is always the positive acute angle between the terminal side and the nearest x-axis.
Correction: Subtract 2π to find a coterminal angle: 7π/3 − 2π = 7π/3 − 6π/3 = π/3. Then evaluate normally: cos(7π/3) = cos(π/3) = 1/2.
Correction: In Quadrant II (π/2 < θ < π), the reference angle is π − θ. For 2π/3: π − 2π/3 = π/3. In Quadrant III (π < θ < 3π/2), the reference angle is θ − π. Never subtract a larger number from a smaller one and accept a negative reference angle.
Fill in all exact values. Leave answers as fractions and radicals. Answers may be placed in the answer key.
| Angle (rad) | Angle (deg) | cos θ | sin θ | tan θ |
|---|---|---|---|---|
0 | 0° | |||
π/6 | 30° | |||
π/4 | 45° | |||
π/3 | 60° | |||
π/2 | 90° | |||
2π/3 | 120° | |||
3π/4 | 135° | |||
5π/6 | 150° | |||
π | 180° | |||
7π/6 | 210° | |||
5π/4 | 225° | |||
4π/3 | 240° | |||
3π/2 | 270° | |||
5π/3 | 300° | |||
7π/4 | 315° | |||
11π/6 | 330° | |||
2π | 360° |
For each angle, name the quadrant and determine whether sin, cos, and tan are positive or negative. Then identify which of the following is possible or impossible (explain your reasoning).
Find the reference angle, state the quadrant, then evaluate all six trig functions exactly.
For each problem, identify what trigonometric concept applies, set up the calculation, and interpret the answer in context.
Each problem contains a student error. Identify the error, explain why it is wrong, and provide the correct solution.
What is wrong? What is the correct answer? (Hint: check the quadrant.)
What is wrong? What is the correct value of cos θ?
Identify the notation error. What does sin⁻¹(1/2) actually mean and what is its value?
Is the computation correct? Is writing −∞ as a value appropriate? What is the correct statement about tan(3π/2)?
What is the error in finding this reference angle? Find the correct reference angle and the quadrant.
Chapter 4 established that as θ increases from 0 to 2π, the point (cos θ, sin θ) travels around the unit circle. The y-coordinate — sine — starts at 0, rises to 1, returns to 0, falls to −1, and returns to 0. The x-coordinate — cosine — starts at 1, falls to 0, continues to −1, rises to 0, and returns to 1. These oscillating values are functions of θ. When plotted with θ on the horizontal axis and the trigonometric value on the vertical axis, they produce continuous wave-shaped graphs.
In Chapter 5, students will graph these waves and analyze their features. Every graph feature connects directly to unit circle behavior:
Students who understand the unit circle thoroughly will find trigonometric graphs intuitive and natural. Students who memorize without understanding will find Chapter 5 confusing. The time invested in Chapter 4 pays forward through every remaining chapter of this book.