The unit circle
Every angle lands on a point of the circle — and that point is (cos θ, sin θ). So drag the angle around and watch sine, cosine and tangent change. The special angles (0°, 30°, 45°, 60°, 90°) even show their exact values.
Tip: drag the point around the circle — the wave trace on the right moves in perfect sync. Switch sin/cos/tan above.
The ideas you're seeing
Ratios are coordinates
On a circle of radius 1, the point at angle θ is exactly (cos θ, sin θ). So the x-coordinate is the cosine and the y-coordinate is the sine.
tan = sin ÷ cos
tan θ = sin θ / cos θ. That's why tan shoots off to infinity at 90° and 270° — there, cos θ = 0 and you can't divide by zero.
The identity
The point always sits on the circle, so sin²θ + cos²θ = 1 for every angle. It's just Pythagoras on the little right triangle.
The special angles
Learn 0°, 30°, 45°, 60°, 90° by heart — they turn up in nearly every exam question. Tap them above to see the exact values.
Part of Priodemy for School — interactive Maths & Science for Class 9–12, free with every EduSuite school. Explore more on the Priodemy for School hub.
Why the unit circle makes trigonometry easier
From triangles to coordinates
Trigonometry is usually introduced with right-angled triangles, where sine is "opposite over hypotenuse". That definition works, but it quietly assumes the angle is between 0° and 90° — a triangle cannot contain an angle of 210°. The unit circle removes that limit. Because the radius is exactly 1, the hypotenuse is 1, so "opposite over hypotenuse" becomes just "opposite", and the two coordinates of the point are the two ratios: x = cos θ and y = sin θ.
Once you accept that, every awkward rule you were asked to memorise turns into something you can simply read off the picture. Drag the angle past 90° and the x-coordinate goes negative — so cosine is negative in the second quadrant. Keep going past 180° and both coordinates are negative. This is the whole content of the "all, sin, tan, cos" rule, except you are deriving it rather than recalling it.
Where tangent comes from, and why it breaks
Tangent is defined as tan θ = sin θ / cos θ, which on the circle is simply y divided by x. That single definition explains the behaviour that otherwise looks arbitrary. As the angle approaches 90°, the x-coordinate shrinks towards zero while y approaches 1, so the ratio grows without bound — tangent is undefined at 90° and 270° because you would be dividing by zero. Watch the value on screen run away as you approach those angles; it is not a bug in the tool, it is the function.
The identity you get for free
The point sits on a circle of radius 1, so by Pythagoras its coordinates satisfy x² + y² = 1. Substituting the definitions gives sin²θ + cos²θ = 1 immediately. This is the most-used identity in the syllabus, and on the unit circle it is not a fact to memorise at all — it is just the equation of the circle, rewritten.
What to try
- Stop at 30°, 45° and 60° and read the exact values. Notice that sin 30° = cos 60° and sin 60° = cos 30° — the complementary-angle relationship, visible as a reflection across the diagonal.
- Compare 30° with 150°: the sine is identical, the cosine has flipped sign. That is the reference-angle idea in one picture.
- Go all the way round past 360° and watch the values repeat. Periodicity stops being an abstract word.
Mistakes that cost marks
Degrees and radians. In Class 11 onward the default unit becomes the radian, where a full turn is 2π rather than 360. Calculus results such as the derivative of sin x being cos x are only true in radians. Setting a calculator to the wrong mode produces confident, completely wrong answers, and it is the single most common source of lost marks in this topic.
Assuming sine and cosine can exceed 1. Because the radius is 1, neither coordinate can be larger than 1 or smaller than −1. If a solution ever gives sin θ = 1.4, the error is upstream — there is no angle that satisfies it. Tangent, being a ratio of the two, has no such limit and takes every real value.