Geometry · Mathematics
Trigonometry Calculator
Solve right and general triangles from sides and angles — enter your knowns, watch the sketch respond, and get every missing dimension in one pass.
Known values(Enter at least two values (sides or angles).)
Results
Right triangle
c² = a² + b²
Enter any two sides or one side plus an acute angle — the diagram updates as you type.
Formulas, solving strategies, and engineering applications
Trigonometry studies relationships between sides and angles of triangles — from Greek trigonon (triangle) and metron (measure). Engineers use it daily when direct measurement is impossible or impractical.
Pythagorean theorem
c² = a² + b²
c = hypotenuse; a, b = legs
Trigonometric ratios
sin(α) = opposite / hypotenuse
cos(α) = adjacent / hypotenuse
tan(α) = opposite / adjacent
Inverse functions
α = arcsin(opposite / hypotenuse)
α = arccos(adjacent / hypotenuse)
α = arctan(opposite / adjacent)
Area
Area = (base × height) ÷ 2
Law of sines
a/sin(A) = b/sin(B) = c/sin(C)
Law of cosines
c² = a² + b² − 2ab·cos(C)
b² = a² + c² − 2ac·cos(B)
a² = b² + c² − 2bc·cos(A)
Area formulas
Area = ½ × base × height
Area = ½ × ab × sin(C)
Area = √(s(s−a)(s−b)(s−c))
Heron's formula; s = (a+b+c)/2
Angle sum
A + B + C = 180°
| Field | Application |
|---|---|
| Structural | Analyze forces in trusses, beams, and supports for stable buildings and bridges. |
| Electrical | Sine and cosine describe AC waveforms and phasor relationships in circuits. |
| Mechanical | Cam profiles, crank mechanisms, and linkage geometry rely on angle–side relations. |
| Civil / survey | Determine distances, elevations, and land areas when direct measurement is impractical. |
Historical origins
Egyptians and Babylonians (~1500 BCE) used triangle ratios for astronomy and construction. Hipparchus (190–120 BCE) is often called the father of trigonometry.
The unit circle
Any point (x, y) on a unit circle equals (cos θ, sin θ), linking geometry to periodic functions.
Euler's identity
e^(iπ) + 1 = 0 ties trigonometry to complex analysis — five fundamental constants in one equation.
Periodic nature
Sin and cos repeat on fixed intervals, making them ideal for waves, tides, and seasonal models.