What is a Trigonometry Calculator?
This calculator computes sin, cos, and tan values, and finds angles using inverse trigonometric functions.
How it's calculated
For a right triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent
Step-by-step example
Right triangle with hypotenuse 10, opposite side 6 — find sinθ and the angle
- Step 1 sinθ = opposite÷hypotenuse = 6÷10 = 0.6
- Step 2 Use the inverse function: θ = arcsin(0.6)
- Step 3 θ ≈ 36.87°
Interpretation: The adjacent side follows from the Pythagorean theorem, √(10²−6²)=8, giving cosθ=0.8 as well.
Why trigonometry was invented
It was created to measure distances you cannot walk to directly — the width of a river, the height of a mountain, the distance to a star. With just one measurable angle and one measurable length, the rest can be calculated. Around 240 BCE, Eratosthenes calculated Earth's circumference using shadow angles measured at the same moment in two cities — accurate to within a few percent. GPS, surveying, and architecture all still rely on this same fundamental principle.
Good to know
- The name 'sine' comes from a translation error. The Sanskrit jya (bowstring) became the Arabic jiba, which, written without vowels, was later misread as jaib (bay or fold) by a Latin translator — producing 'sinus' (bay), the root of 'sine.' A bowstring somehow became a bay.
- 'Tangent' literally means 'touching' — the tangent value equals the length of the line segment tangent to a unit circle.
- Trigonometric functions describe waves themselves — sound, light, and electrical signals can all be expressed as sums of sine waves through the Fourier transform, the mathematical foundation behind MP3 and JPEG compression.
- This links math directly to music — concert pitch A vibrates at 440Hz, and chords sound pleasant when frequency ratios form simple whole-number relationships.
- sin²θ + cos²θ = 1 is really just the Pythagorean theorem divided through by the hypotenuse squared — not a new fact, just the same relationship expressed differently.
Frequently asked questions
Q. Should angles be entered in degrees or radians?
Depends on the calculator setting. Calculus and physics typically use radians; everyday surveying uses degrees.
Q. Why is tan(90°) undefined?
Because the adjacent side becomes zero, which would require dividing by zero — mathematically undefined.
Q. What are inverse trigonometric functions?
Functions that work backward from a ratio to find the angle, written as arcsin, arccos, and arctan.