Explore our all-in-one interactive unit circle chart. Master angles, radians, sine, cosine, and tangent with ease. Our visual tool makes trigonometry simple, providing a complete unit circle table, diagram, and reference for all your needs.
🖱️ Interactive Learning: Drag the point around the circle to explore different angles and see how sine, cosine, and tangent values change in real-time.
⚙️ Display Options: Use the control panel to show/hide sine, cosine, tangent lines, coordinates, and other helpful visual elements.
📐 Special Angles: Enable "Snap to Special Angles" to automatically lock onto important angles like 30°, 45°, 60°, etc.
🔢 Units: Switch between degrees and radians to see angles in both measurement systems.
The "Unit Circle" is a circle with a radius of exactly 1, centered at the origin (0,0) of a graph.
Its simplicity is its power, making it the perfect tool for understanding how angles and trigonometric functions relate to the x and y coordinates of a point.
For any point on the edge of the circle, the distance from the center to that point is always 1. We represent this point with coordinates (x, y).
These coordinates are the key to understanding trigonometry.
Since the radius of the unit circle is 1, we can directly read the sine, cosine, and tangent values from the coordinates and line segments:
This direct relationship makes the unit circle incredibly powerful for understanding trigonometry. Instead of memorizing abstract ratios, you can visually see the function values as coordinates and line lengths. Our interactive tool visualizes this by showing:
Use the interactive tool above to drag the point to 0°. You'll see that cos(0°) = 1 and sin(0°) = 0, which matches the coordinates (1, 0)!
Pythagoras' Theorem says that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:
x² + y² = 1²
But 1² is just 1, so we get the equation of the unit circle:
x² + y² = 1
Also, since the coordinates are given by the trigonometric functions x = cos(θ) and y = sin(θ), we can substitute them into the equation:
(cos(θ))² + (sin(θ))² = 1
This is a very useful "identity" in trigonometry.
The unit circle is divided into four quadrants, and the signs of sine, cosine, and tangent depend on which quadrant the angle falls in:
| Quadrant | sin | cos | tan |
|---|---|---|---|
| I (0° to 90°) | + | + | + |
| II (90° to 180°) | + | − | − |
| III (180° to 270°) | − | − | + |
| IV (270° to 360°) | − | + | − |
Why these signs? Remember that cosine = x-coordinate and sine = y-coordinate. In Quadrant II, x is negative but y is positive, so cos < 0 and sin > 0.
While the unit circle helps us understand angles of any measure, you should try to remember the sine, cosine, and tangent values for the most common angles. This unit circle table of values forms the foundation of trigonometry.
| Angle (degrees) | Angle (radians) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | -1 | 0 |
| 270° | 3π/2 | -1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Memorizing trigonometric values becomes much easier with the right tricks. Here are the most effective methods:
This pattern is a great way to remember the values for sine and cosine. By drawing lines from the points on the unit circle to the axes, you can see a clear pattern:
Tip: You only need to remember the sequence 1, 2, 3 for the numbers inside the square roots!
You only need to memorize these three fractions:
These can be arranged to give you all the sine and cosine values for 30°, 45°, and 60°.
This is a powerful trick to remember the trig values for key angles. Imagine the angles 0°, 30°, 45°, 60°, and 90° as 5 points on an arc.
To find the sine or cosine, you just need to count the dots above or below a specific point, as shown in the diagram. The diagram on the right contains the formulas and a visual example for 30°.
For a 45° angle, the (x, y) coordinates are equal, so we have a simple case where x = y. Substituting this into the unit circle equation (x² + y² = 1):
So, at 45°, the coordinates are (√2/2, √2/2).
For 60°, we can imagine an equilateral triangle with sides of length 1 placed at the origin. When we split it in half, we get a 30-60-90 triangle.
From the diagram, we can see that the x-coordinate (cos 60°) is 1/2.
We can find the y-coordinate using the unit circle equation:
For 30°, the x and y values are simply swapped.
These examples show how we can derive exact values for special angles using basic geometry. The same principles can be applied to find values for 120°, 135°, 150°, and other angles by using symmetry properties of the unit circle.
Here's a complete overview of all 16 commonly used special angles around the unit circle, often called a unit circle chart or diagram. Notice the symmetry patterns:
Use the interactive tool above to explore these angles. Try dragging the point to each of these special positions and observe how the values match what you've learned. Pay attention to the symmetry patterns between quadrants!
The unit circle is a fundamental tool in trigonometry used to visualize and understand the relationships between angles and trigonometric functions. It helps in determining the sine, cosine, and tangent values of any angle, understanding function periodicity, and deriving trigonometric identities.
For any angle θ, the point on the circle is (x, y). The x-coordinate is the cosine of the angle (cos(θ)), and the y-coordinate is the sine of the angle (sin(θ)).
A radius of 1 simplifies the definitions of sine and cosine. In a right triangle inside the circle, the hypotenuse is always 1, so sin(θ) = opposite/1 = opposite and cos(θ) = adjacent/1 = adjacent. This makes the coordinates directly equal to the function values.
Degrees and radians are two different units for measuring angles. A full circle is 360° or 2π radians. To convert from degrees to radians, multiply by π/180. To convert from radians to degrees, multiply by 180/π.
You don't have to memorize the entire circle! The key is understanding reference angles. Think of a reference angle as the "mirror image" of an angle in the first quadrant. It's the smallest acute angle formed between the terminal arm and the x-axis. For instance, if you're asked "what is the reference angle for 150 degrees?", you'll find it's 30° because 150° is 30° away from the x-axis (180°). This means the absolute trig values for 150° are the same as for 30°. Simply use the quadrant rule (ASTC) to determine the correct sign for calculating sine and cosine for any angle.
It's like running laps on a circular track. After one full lap (360°), you're back at the start. So, for finding trig values for angles greater than 360, like 400°, you just subtract 360°. The angle 400° ends at the exact same spot as 40°, so they have the same trig values. When you need to know what to do with negative angles on the unit circle, like -60°, just move clockwise instead of counter-clockwise. You'll land on the same spot as 300°. These are called coterminal angles.
The simplest intuitive way to understand tangent on the unit circle is to think of it as the slope of the line from the origin to the point on the circle. A steeper line means a larger tangent value. For a more precise visualization, imagine a vertical line tangent to the circle at (1, 0). The y-value where the radius line (or its extension) intersects this tangent line is the value of tan(θ). This also explains why is tangent undefined at 90 degrees: the radius line becomes vertical and parallel to the tangent line, so they never intersect!
The periodicity of sine and cosine explained simply comes back to the circular track analogy. Every time you complete a full 360° rotation, you are back at the same (x, y) point on the circle. Since cos(θ) = x and sin(θ) = y, the function values must repeat exactly. This predictable, repeating pattern is called "periodicity," and it's a fundamental property you need for understanding the period of trigonometric functions.