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The Unit Circle

The circle of radius 1 where the point at angle θ is (cos θ, sin θ). The four quadrant sign rules, and the exact values at 0, 30, 45, 60 and 90 degrees.

Unit circle

The unit circle is the circle of radius 1 centred on the origin, on which the point at angle θ has coordinates (cos θ, sin θ).

طلبہ کہاں پڑھتے ہیں: Grade 9 or 10, at the point where sine and cosine have to work for angles beyond 90°, and again throughout Digital SAT trigonometry.

مختصر جواب
The unit circle is the circle of radius 1 centred on the origin. Rotate anticlockwise from the positive x-axis through an angle θ and the point you land on has coordinates (cos θ, sin θ). That one fact defines sine and cosine for every angle, not only for the acute angles inside a triangle.

ایک مثال

150° → (−√3/2, 1/2), so cos 150° = −√3/2 and sin 150° = 1/2

Rotating 150° anticlockwise lands in the second quadrant, where x is negative and y is positive. The point sits 30° short of the negative x-axis, so it is the mirror image of the 30° point: the same distances from the axes, 1/2 up and √3/2 across, but with the x coordinate now negative. Reading the coordinates off gives both values at once.

Coordinates, not triangle ratios

SOHCAHTOA defines sine and cosine as ratios of sides in a right-angled triangle, which quietly caps them at angles under 90°: there is no triangle with a 120° angle in it that has a hypotenuse to divide by. The unit circle replaces the ratio with a position, and a position exists at any angle at all.

The link between the two is that the radius is 1. Drop a vertical from the point on the circle to the x-axis and you have a right-angled triangle with hypotenuse 1, so cos θ = adjacent ÷ 1 is just the x coordinate, and sin θ = opposite ÷ 1 is just the y coordinate. Nothing new has been invented; the same numbers have been given somewhere to live.

One identity falls straight out of this. Every point on the circle satisfies x² + y² = 1, and since x is cos θ and y is sin θ, cos²θ + sin²θ = 1 for every angle. It is Pythagoras' theorem on a triangle with hypotenuse 1, which is why it is called the Pythagorean identity.

The signs in the four quadrants

Because cosine is the x coordinate and sine is the y coordinate, the sign of each is decided by nothing more than which quadrant the point is in. There is no rule to memorise beyond the axes you already know from coordinate geometry, and tangent, being y ÷ x, is positive wherever x and y share a sign.

Textbooks often teach this as a mnemonic — All, Sine, Tangent, Cosine going anticlockwise from the first quadrant, marking which ratio is positive where. The mnemonic is a compressed version of the table below, and it is worth knowing only if you can also rebuild it. A student who sketches the circle will never need it; a student who has only the mnemonic has nothing to fall back on when they misremember the order.

First quadrant, 0° to 90°: cos +, sin +, tan +
Second quadrant, 90° to 180°: cos −, sin +, tan −
Third quadrant, 180° to 270°: cos −, sin −, tan +
Fourth quadrant, 270° to 360°: cos +, sin −, tan −

The five angles worth knowing exactly

Five points on the circle carry exact values that examiners expect without a calculator, and they are easier to hold as coordinates than as a table of surds. Notice that the x coordinates run 1, √3/2, √2/2, 1/2, 0 while the y coordinates run the same list backwards — the pattern is symmetric about 45°, where the point sits on the line y = x.

The values themselves come from two triangles rather than from memory. A square cut along its diagonal gives an isosceles right-angled triangle and the 45° values; an equilateral triangle cut down the middle gives a 30-60-90 triangle and the other two. Any of the five can be rebuilt from those two pictures in under a minute, which is the safety net worth having.

0°: (1, 0) — cos 0° = 1, sin 0° = 0
30°: (√3/2, 1/2) — tan 30° = 1/√3
45°: (√2/2, √2/2) — tan 45° = 1
60°: (1/2, √3/2) — tan 60° = √3
90°: (0, 1) — tan 90° is undefined, because x = 0

عام سوالات

Why is it called the unit circle?

Because its radius is one unit. That choice is what makes the coordinates equal to the cosine and sine themselves rather than to r cos θ and r sin θ. On a circle of radius 5 the point at angle θ is (5 cos θ, 5 sin θ), which works perfectly well but carries a factor around that adds nothing.

Do I need the unit circle for GCSE maths?

Not by name. GCSE and IGCSE papers handle angles beyond 90° through the shapes of the sine and cosine graphs instead. The unit circle is worth meeting anyway, because it explains why those graphs have the symmetry they do — the graph is what you get by unrolling the circle along an axis.

Which way round do the angles go?

Anticlockwise from the positive x-axis is the positive direction, so 90° is straight up. A clockwise rotation is written as a negative angle, which is why cos(−θ) = cos θ and sin(−θ) = −sin θ: reflecting the point in the x-axis leaves its x coordinate alone and flips the sign of its y coordinate.

How does the unit circle help on the Digital SAT?

SAT trigonometry questions mix degrees and radians and often ask for a value at an angle outside the first quadrant. Reading the answer off a sketched circle is faster and less error-prone than recalling a sign rule, and the same sketch settles which of two answer choices has the right sign.

حوالہ جات

The Math Section — SAT Suite of Assessments — College Board

آخری تازہ کاری 21 اگست، 2026