What the Unit Circle Is
The unit circle is a circle of radius 1 centred at the origin. Walk an angle θ counter-clockwise from the positive x-axis and you land on the point (cos θ, sin θ). That is the whole idea: cosine is the x-coordinate and sine is the y-coordinate.
Because the radius is 1, the point also satisfies x² + y² = 1, which is the identity sin²θ + cos²θ = 1 in disguise. Tangent is the slope of the line from the origin to the point, so tan θ = sin θ ÷ cos θ.
The Special Angles You Should Know
| Degrees | 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 |
Notice the pattern in the sine column going down: √0/2, √1/2, √2/2, √3/2, √4/2. Cosine is the same list in reverse. Learn the first quadrant and the other three are reflections of it.
How to Use a Reference Angle
Every angle has a reference angle: the acute angle it makes with the x-axis. Find it, read the first-quadrant value, then fix the signs by quadrant. All three ratios are positive in quadrant I, only sine in II, only tangent in III and only cosine in IV.
sin 225°, cos 225°, tan 225°
- 225° is between 180° and 270°, so it lies in quadrant III.
- Reference angle: 225° − 180° = 45°.
- At 45°, cos = sin = √2/2 and tan = 1.
- In quadrant III sine and cosine are negative and tangent is positive.
- So cos 225° = −√2/2, sin 225° = −√2/2 and tan 225° = 1.
Answer: (−√2/2, −√2/2), tan = 1
Converting Between Radians and Degrees
A half turn is 180° and also π radians, so multiply degrees by π/180 to get radians and radians by 180/π to get degrees.
7π/6 in degrees, then (cos, sin)
- Convert: 7π/6 × 180/π = 210°.
- 210° is in quadrant III, because it is between 180° and 270°.
- Reference angle: 210° − 180° = 30°.
- At 30°, cos = √3/2 and sin = 1/2. Both are negative in quadrant III.
Answer: (−√3/2, −1/2)
The most common calculator mistake is being in the wrong mode. sin(30) is 0.5 in degree mode but about −0.988 in radian mode. If your answer looks strange, check the mode before anything else.
A Hand Trick for the First Quadrant
Hold up your left hand, palm facing you. Let the thumb be 0° and the little finger 90°, with the fingers between them being 30°, 45° and 60°. To read sine at an angle, count the fingers below that finger and put the number under a root over 2. At 60° there are three fingers below it, so sin 60° = √3/2. For cosine, count the fingers above.
The Values to Know
| Angle | Radians | sin | cos |
|---|---|---|---|
| 0° | 0 | 0 | 1 |
| 30° | π/6 | 1/2 | √3/2 ≈ 0.8660 |
| 45° | π/4 | √2/2 ≈ 0.7071 | √2/2 ≈ 0.7071 |
| 60° | π/3 | √3/2 ≈ 0.8660 | 1/2 |
| 90° | π/2 | 1 | 0 |
Everything else follows by symmetry. 150° has the same sine as 30° and the cosine changes sign, because 150° is in quadrant II. The sine pattern 0, 1/2, √2/2, √3/2, 1 comes from √0/2, √1/2, √2/2, √3/2, √4/2, an easy memory trick.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: memorising the whole circle. You need one quadrant plus the sign rule.
Underrated: reference angles. Reduce any angle to 0–90° and then fix the sign by the quadrant.
Plain wrong: forgetting that the point is (cos θ, sin θ), so x is cosine and y is sine, not the other way round.
Unit Circle Mistakes
- Mixing up which coordinate is sine. It is the y-coordinate; cosine is x.
- Forgetting that tan 90° and tan 270° are undefined, because cosine is zero there.
- Using the reference angle but not changing the sign for the quadrant.
- Stopping at 360° when the angle is bigger. Subtract full turns first: 405° behaves like 45°.
- Leaving a radical in the denominator. Write √2/2, not 1/√2.
Why Students Get Stuck on The Unit Circle, and What a Good Calculator Should Do
Searching for help with the Unit Circle usually starts late at night with one problem that will not work out. It has to be memorised, and the signs by quadrant are easy to forget. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to read exact values and understand the pattern, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Forgetting the sign of sine and cosine in each quadrantsin(pi/6)↗
- Mixing up degrees and radianscos(5*pi/4)↗
- Not knowing the reference angletan(pi/3)↗
- Swapping the sine and cosine valuessin(150 degrees)↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants exact values, not decimals
- Wants the reference angle shown
- Wants degrees and radians together
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.sin(pi/6)↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.