How this instrument works
Most angles' sine and cosine values are irrational numbers with no clean closed form — but five special angles, 0°, 30°, 45°, 60°, and 90°, have EXACT values derivable directly from two simple triangles: a 30-60-90 triangle (built from an equilateral triangle split in half) and a 45-45-90 triangle (an isosceles right triangle). These five angles recur constantly across trigonometry, geometry, and calculus, which is why memorizing their exact values outright pays off far more than looking them up every time.
The 30-60-90 triangle's fixed side ratios (1, √3, and 2) give exact values for 30° and 60° directly; the 45-45-90 triangle's matching legs give the shared value √2⁄2 for both sine and cosine at 45°. The boundary angles, 0° and 90°, are simpler still — one of the two functions reaches its maximum value of 1 while the other drops to exactly 0.
This site's other trigonometry pages compute sine and cosine for ANY angle, typically returning a rounded decimal approximation; this page instead restricts to just these five angles specifically so the EXACT, non-decimal answer can be shown directly.
- Select one of the five special angles: 0°, 30°, 45°, 60°, or 90°.
- Read sin(angle) and cos(angle) — both given as exact values, not decimal approximations.
- Compare 30° against 60° to see how sine and cosine swap roles between complementary angles.
- Notice that 45° gives sine and cosine the identical value — the defining symmetry of an isosceles right triangle.
Worked example — 30° and 45°
At 30°, sine and cosine take exact values derivable from a 30-60-90 triangle's own fixed side ratios (1, √3, 2): sin(30°)=1⁄2 exactly, and cos(30°)=√3⁄2 — values worth memorizing outright, since they recur constantly and are exact rather than rounded approximations.
At 45°, sine and cosine are equal to each other, √2⁄2≈0.707 — the defining symmetry of an isosceles right triangle, where the two legs match exactly. At 90°, sine reaches its maximum value, 1, while cosine drops to exactly 0 — the point where a right triangle's own angle would have flattened out entirely.
Questions
Why do only five angles have exact trig values worth memorizing?
Because they're the only angles between 0° and 90° derivable directly from two simple, well-known triangles — a 30-60-90 triangle and a 45-45-90 triangle — whose fixed side ratios give clean, non-decimal results; every other angle's sine and cosine are irrational with no comparably simple closed form.
Where do the 30-60-90 triangle's exact values come from?
From splitting an equilateral triangle exactly in half — the resulting right triangle has fixed side lengths in the ratio 1 to √3 to 2, giving sin(30°)=1⁄2 and cos(30°)=√3⁄2 directly from those ratios.
Why are sine and cosine equal at 45°?
Because 45° comes from an isosceles right triangle, where the two legs are exactly equal in length — sine and cosine each compare one leg to the hypotenuse, and with matching legs, both ratios come out identical.
How is this different from this site's other trigonometry calculators?
Those pages accept ANY angle and typically return a rounded decimal approximation; this page restricts to just these five special angles specifically so the exact, non-decimal value can be shown directly instead.
Why do sine and cosine swap between 30° and 60°?
Because 30° and 60° are complementary angles (they add to 90°), and sine of one angle always equals cosine of its complement — a general trigonometric identity, visible directly here since sin(30°)=cos(60°)=1⁄2.