Unit Circle Chart with Trig Values

How the Unit Circle Works

The Core Idea

Draw a circle with radius 1 on a coordinate plane, centered at (0, 0). Pick any angle theta measured counter-clockwise from the positive x-axis. The point where the angle's ray hits the circle is at coordinates:

That is it. The x-coordinate is cosine, and the y-coordinate is sine. Because the radius is 1, no scaling is needed.

Why Radius = 1 Matters

On a circle of radius r, the point is (r * cos theta, r * sin theta). Setting r = 1 removes the multiplier, so trig values read directly from the coordinates. This is why it is called the unit circle.

All Six Trig Functions

The reciprocal functions (csc, sec, cot) are undefined wherever their denominator is zero.

Quadrant Sign Rules

The mnemonic "All Students Take Calculus" tells you which trig functions are positive in each quadrant:

Reference Angles

A reference angle is the acute angle between the terminal side and the x-axis. It lets you reuse values from Quadrant I in other quadrants -- just apply the correct sign. For example, sin 150 has the same absolute value as sin 30 (both 1/2), but you do not negate because 150 is in Quadrant II where sine is positive.

Degrees to Radians Conversion

Common conversions: 30 = pi/6, 45 = pi/4, 60 = pi/3, 90 = pi/2, 180 = pi, 360 = 2pi.

The Three Key Triangles

Most unit circle values come from three right triangles inscribed in the circle:

• 30-60-90 triangle: sides 1, sqrt(3)/2, 1/2 -- gives values for 30 and 60 degrees
• 45-45-90 triangle: sides 1, sqrt(2)/2, sqrt(2)/2 -- gives values for 45 degrees
• Axis points: 0, 90, 180, 270 land on the axes with coordinates of 0 and +/-1

Every other standard angle is a reflection of one of these into another quadrant.

Common Mistakes

• Mixing up sine and cosine: Sine = y-coordinate (vertical), cosine = x-coordinate (horizontal). A common trick: "x comes before y alphabetically, cosine comes before sine alphabetically."
• Forgetting signs in Q3 and Q4: Both coordinates are negative in Quadrant III. Only x is positive in Quadrant IV.
• Radians vs. degrees in a calculator: If your calculator says sin(90) is about 0.894, it is in radian mode. Switch to degree mode, or use sin(pi/2).
• Tangent at 90 and 270: Tangent is undefined (not zero) at these angles, because cos theta = 0.
• Confusing sqrt(2)/2 with sqrt(3)/2: At 45 both sine and cosine are sqrt(2)/2. At 30 and 60 you get one of each (sqrt(3)/2 and 1/2).

Worked Examples

Example 1: Find sin, cos, and tan of 150

Step 1 -- Find the reference angle. 150 is in Quadrant II. The reference angle is 180 - 150 = 30.

Step 2 -- Look up the Quadrant I values for 30. sin 30 = 1/2, cos 30 = sqrt(3)/2, tan 30 = sqrt(3)/3.

Step 3 -- Apply quadrant signs. In Quadrant II, sine is positive, cosine and tangent are negative.

Result:

• sin 150 = 1/2 (approx 0.5000)
• cos 150 = -sqrt(3)/2 (approx -0.8660)
• tan 150 = -sqrt(3)/3 (approx -0.5774)

The coordinates on the unit circle at 150 are (-sqrt(3)/2, 1/2).

Example 2: Find sin, cos, and tan of 315

Step 1 -- Find the reference angle. 315 is in Quadrant IV. The reference angle is 360 - 315 = 45.

Step 2 -- Look up the Quadrant I values for 45. sin 45 = sqrt(2)/2, cos 45 = sqrt(2)/2, tan 45 = 1.

Step 3 -- Apply quadrant signs. In Quadrant IV, cosine is positive, sine and tangent are negative.

Result:

• sin 315 = -sqrt(2)/2 (approx -0.7071)
• cos 315 = sqrt(2)/2 (approx 0.7071)
• tan 315 = -1

The coordinates on the unit circle at 315 (7pi/4 radians) are (sqrt(2)/2, -sqrt(2)/2).

Example 3: Convert 5pi/6 radians to degrees and find trig values

Step 1 -- Convert to degrees. 5pi/6 x 180/pi = 150.

Step 2 -- This is the same as Example 1. sin(5pi/6) = 1/2, cos(5pi/6) = -sqrt(3)/2, tan(5pi/6) = -sqrt(3)/3.

This shows why memorizing radians-to-degrees conversions pays off -- the same reference angle logic applies either way.

Privacy and Limitations

This unit circle chart runs entirely in your browser. No data is sent to any server, and no information is stored or tracked. The tool covers the 16 standard angles (multiples of 30 and 45 degrees). For non-standard angles, the tool computes approximate decimal values using JavaScript's built-in trig functions. This is a reference and learning tool -- it does not solve equations or prove identities.

Frequently Asked Questions

What is the unit circle?

The unit circle is a circle with radius 1 centered at the origin (0, 0) of a coordinate plane. For any angle theta measured from the positive x-axis, the point on the circle has coordinates (cos theta, sin theta). It is the fundamental tool for defining sine, cosine, and tangent for all angles -- not just those in a right triangle.

How do you find sine and cosine on the unit circle?

Draw the angle from the positive x-axis. Where the terminal side intersects the circle, the x-coordinate is cosine and the y-coordinate is sine. No formula needed beyond reading coordinates.

What are the exact values at 30, 45, and 60 degrees?

At 30 (pi/6): sin = 1/2, cos = sqrt(3)/2. At 45 (pi/4): sin = sqrt(2)/2, cos = sqrt(2)/2. At 60 (pi/3): sin = sqrt(3)/2, cos = 1/2. These three reference angles produce all non-axis values on the standard unit circle.

How do you convert degrees to radians?

Multiply degrees by pi/180. Example: 90 x pi/180 = pi/2 radians. To convert back, multiply radians by 180/pi.

What is tangent on the unit circle?

Tangent equals sin theta divided by cos theta, or equivalently the y-coordinate divided by the x-coordinate. It is undefined at 90 and 270 because cos theta = 0 at those angles.

Why is it called the unit circle?

"Unit" means 1. The radius is exactly 1 unit, which makes coordinates equal trig values directly.

What are the quadrant signs for trig functions?

Q1: all positive. Q2: sine positive. Q3: tangent positive. Q4: cosine positive. Remember: "All Students Take Calculus."

How many radians are in a full circle?

2pi radians = 360 degrees. A semicircle is pi radians (180 degrees). A quarter-turn is pi/2 radians (90 degrees).

Does the unit circle work for angles greater than 360 degrees?

Yes. Angles wrap around. 390 degrees has the same trig values as 30, because 390 - 360 = 30. Negative angles rotate clockwise: -90 is the same position as 270.

How is the unit circle used in real life?

Anywhere something rotates or oscillates: physics (waves, circular motion), engineering (signal processing, AC circuits), computer graphics (rotation matrices), music (sound waves), and navigation (bearings and headings).

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