TAHA Engineering Toolsby TAHA Knowledge Lab

Mathematics & Statistics

Trigonometry Calculator

Six trig functions of an angle, inverse sin/cos/tan with all solutions, and a right-triangle solver.

Inputs

Calculate from
Unit circle with an angle theta from the positive x-axis; the point on the circle has horizontal coordinate cos theta and vertical coordinate sin theta.θsin θcos θ(cos θ, sin θ)1IIIIIIIV
On the unit circle (radius 1), the point at angle θ has coordinates (cos θ, sin θ); tan θ = sin θ ÷ cos θ. The signs follow the quadrants I to IV. Diagram not to scale.

Result

sin θ
0.5
cos θ
0.866025403784
tan θ
0.57735026919
csc θ = 1 / sin θ
2
sec θ = 1 / cos θ
1.15470053838
cot θ = 1 / tan θ
1.73205080757
Position
Quadrant I (30° from 0° to 360°); reference angle 30°
Working
θ = 30° = 0.5235987756 rad sin θ = 0.5, cos θ = 0.8660254038, tan θ = sin ÷ cos = 0.5773502692

For θ = 30 °: sin θ = 0.5, cos θ = 0.866025403784, tan θ = 0.57735026919.

Educational estimate — check the assumptions below.

About this calculator

Trigonometry links angles and lengths. In a right triangle, “SOH CAH TOA” gives sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse and tan = opposite ÷ adjacent. On the unit circle the same functions are defined for any angle. This calculator gives all six functions of an angle, the angle for a given sine, cosine or tangent (with every solution from 0° to 360°), and solves a right triangle from two known values — useful for slopes, roof pitches, ramps, taper angles and setting-out.

Formula

Functions of an angle

sin θ = y, cos θ = x on the unit circle    tan θ = sin θ ÷ cos θ

csc θ = 1/sin θ   sec θ = 1/cos θ   cot θ = 1/tan θ    sin² θ + cos² θ = 1

Inverse (angle from value)

θ = arcsin v (−90° … 90°)   θ = arccos v (0° … 180°)   θ = arctan v (−90° … 90°)

Other solutions: sin θ = sin(180° − θ); cos θ = cos(360° − θ); tan θ = tan(θ + 180°).

Right-triangle solver

c² = a² + b²    sin A = a/c   cos A = b/c   tan A = a/b    A + B = 90°

Angle C is the right angle; leg a is opposite angle A and leg b is next to it. A leg from the hypotenuse is computed as √((c − a)(c + a)) to avoid rounding loss.

Worked example

  • θ = 30°: sin = 0.5, cos = 0.866025403784, tan = 0.57735026919
  • sin θ = 0.5 → θ = 30°; all solutions 30° and 150°
  • Legs a = 3 and b = 4: c = √(9 + 16) = 5, A = arctan(3/4) = 36.8699°, B = 53.1301°, area 6, perimeter 12

Assumptions and limitations

  • Angles can be entered and shown in degrees, radians, gradians or turns (the unit menus).
  • At exact multiples of 90° the calculator returns exact values (sin 180° = 0, cos 90° = 0) and reports tan, sec, csc or cot as undefined where they are.
  • The triangle solver is for right triangles only (C = 90°). Lengths are in any one unit; the area is in that unit squared.

References

  • Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, 4.3 (circular functions).
  • Kreyszig, E. Advanced Engineering Mathematics — trigonometric identities.