Gera Tools

Trigonometric Values Reference

sin, cos, tan values at 0, 30, 45, 60, 90 degrees and key angles.

Reference table of exact trigonometric values (sin, cos, tan) at standard angles in degrees and radians, plus a live evaluator that computes sin, cos, and tan for any angle you enter. Runs entirely in your browser. It runs free in your browser on Gera Tools, with nothing uploaded.

Last updated Source: Gera Tools

What are the exact values at the special angles?

At 30 degrees sin is 1/2 and cos is sqrt(3)/2; at 45 degrees both sin and cos are sqrt(2)/2; at 60 degrees they swap to sqrt(3)/2 and 1/2. These exact surd forms come from the 30-60-90 and 45-45-90 right triangles.

Trigonometry rests on a handful of exact values at special angles that show up everywhere from geometry homework to graphics code. This reference lists sin, cos, and tan at the standard angles in both degrees and radians, and adds a live evaluator so you can compute the three ratios for any angle you type.

How it works

The standard table holds exact symbolic values derived from two special right triangles: the 45-45-90 triangle gives sin 45 = cos 45 = √2/2, and the 30-60-90 triangle gives the 1/2 and √3/2 pair at 30 and 60 degrees. On the unit circle, a point at angle θ sits at coordinates (cos θ, sin θ), and tangent is their ratio:

tan(θ) = sin(θ) / cos(θ)

The evaluator first converts your angle to radians using radians = degrees × π/180, then calls the standard sine and cosine routines. Where cosine is effectively zero — at 90 and 270 degrees — tangent is reported as undefined rather than a huge or misleading number, matching its true vertical asymptote.

Tips and example

  • Memorize the 45-degree row first: sin = cos = √2/2 ≈ 0.7071, tan = 1. The 30 and 60 rows are mirror images of each other.
  • Past 90 degrees the signs flip by quadrant: in the second quadrant sin stays positive while cos and tan go negative, which the extended table (120, 135, 150 degrees) shows directly.
  • For radian input, remember π ≈ 3.14159, so entering 1.5708 (≈ π/2) gives sin ≈ 1, cos ≈ 0.

The evaluator is handy for spot-checking a formula or confirming an angle conversion without reaching for a separate calculator.

The standard angle table

DegreesRadianssincostan
00010
30π/61/2√3/21/√3
45π/4√2/2√2/21
60π/3√3/21/2√3
90π/210undefined
1202π/3√3/2−1/2−√3
1353π/4√2/2−√2/2−1
1505π/61/2−√3/2−1/√3
180π0−10
2703π/2−10undefined
3602π010

Where these values come from

The exact values at 30, 45, and 60 degrees come from two constructions:

The 45-45-90 triangle: a right isosceles triangle with legs of length 1 has a hypotenuse of √2. Dividing each side by √2 normalises it to a hypotenuse of 1, giving sin 45 = cos 45 = 1/√2 = √2/2.

The 30-60-90 triangle: start with an equilateral triangle of side 2 and drop an altitude to bisect it. The half-triangle has sides 1, √3, and 2, giving sin 30 = 1/2, cos 30 = √3/2, sin 60 = √3/2, cos 60 = 1/2.

Beyond 90 degrees the signs flip by quadrant: ASTC (All, Sine, Tangent, Cosine) or the mnemonic “All Students Take Calculus” tells you which functions stay positive in each quadrant.

Practical uses

  • Graphics and game development: rotating a vector by angle θ maps (x, y) to (x cos θ − y sin θ, x sin θ + y cos θ). The exact values at 90-degree multiples simplify these to sign flips without floating-point error.
  • Physics problems: resolving a force at a known angle into horizontal and vertical components uses sin and cos directly. A 30-degree incline with force F has a vertical component F sin 30 = F/2.
  • Exam shortcuts: knowing exact surd values lets you leave answers in exact form (e.g. √3/2) rather than a decimal approximation, which is often required in A-level and university maths.