Trigonometry Calculator
Trig values for any angle, inverse functions, unit circle and triangle solver
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Key Identities
sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
Law of Sines: a/sin A = b/sin B = c/sin C
Law of Cosines: c² = a² + b² − 2ab·cos C
Area: ½ × a × b × sin C
Gradians: 400 grad = 360° = 2π rad
Angle
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In Degrees / Radians / Gradians
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sin θ
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cos θ
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tan θ
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cot θ
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sec θ
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csc θ
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Inverse Trig (returns angle in degrees)
arcsin(sin θ)
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arccos(cos θ)
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arctan(tan θ)
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Unit Circle
90°
270°
0°
180°
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What is a Trigonometry Calculator?
Trigonometry deals with relationships between angles and sides of triangles. The six fundamental functions — sin, cos, tan, csc, sec, cot — are used across physics, engineering, navigation, architecture, and computer graphics. This calculator evaluates all six functions in both degrees and radians.
The triangle solver mode uses the Law of Sines and Law of Cosines to find unknown sides and angles from any valid combination of known values (SSS, SAS, ASA, AAS, SSA) — ideal for geometry problems and real-world triangle calculations.
lightbulb Example Calculation
Scenario: Arjun, Class 11 JEE aspirant from Kota — needs all 6 trig values at 30° for physics problems involving inclined planes and projectile motion (his Resonance batch test is tomorrow)
1sin(30°) = 0.5 | cos(30°) = 0.866 | tan(30°) = 0.577
2csc(30°) = 2 | sec(30°) = 1.155 | cot(30°) = 1.732
3In radians: 30° = π/6 ≈ 0.5236 rad
✓ All 6 trig functions computed for 30°
help_outlineHow to Use the Trigonometry Calculator
- Select the Mode — "Angle → Trig Values" to find all 6 trig functions for a given angle, or "Triangle Solver" to find missing sides and angles of any triangle.
- In Angle mode: enter the angle value and select Degrees, Radians, or Gradians — all three formats are supported and converted automatically.
- Click "Calculate" — sin, cos, tan, cot, sec, and csc values appear along with inverse trig results; the unit circle visual shows the angle's position.
- In Triangle Solver mode: select the known case (SSS, SAS, ASA, AAS, or SSA) and enter the known sides (a, b, c) and/or angles (A, B, C in degrees).
- Click "Solve Triangle" — all unknown sides, angles, the triangle area, and perimeter are computed; the triangle type (acute, obtuse, right) is identified.
Benefits
- All 6 trig functions (sin, cos, tan, cot, sec, csc) computed simultaneously — no need for separate calculations
- Supports Degrees, Radians, and Gradians — converts between all three automatically
- Triangle solver handles all 5 congruence cases including the ambiguous SSA case
- Visual unit circle shows angle position — helps understand quadrant behavior and sign of functions
- Inverse trig values (arcsin, arccos, arctan) shown alongside forward functions for complete reference
Key Terms
- Sine (sin θ)
- Opposite ÷ Hypotenuse in a right triangle; ranges from −1 to 1; reaches maximum (1) at 90°
- Cosine (cos θ)
- Adjacent ÷ Hypotenuse; equals 1 at 0°, 0 at 90°; used extensively in dot products and wave equations
- Tangent (tan θ)
- sin θ ÷ cos θ = Opposite ÷ Adjacent; undefined at 90° and 270° where cos = 0
- Radian
- SI unit of angle: 2π rad = 360°; 1 rad ≈ 57.3°. All calculus formulas (derivatives of sin, cos) require radians
- Law of Cosines
- c² = a² + b² − 2ab·cos C — generalises the Pythagorean theorem; used for SAS and SSS triangle cases
quizFrequently Asked Questions
When should I use degrees vs radians?
Use degrees for everyday geometry, navigation, and problems that explicitly state angle in degrees — it's more intuitive (full circle = 360°). Use radians for calculus and physics: the derivative of sin(x) = cos(x) is only true when x is in radians. Programming languages like Python's math.sin() and JavaScript's Math.sin() use radians by default. For JEE Physics, most problem statements give angles in degrees, but SHM, wave, and rotational dynamics formulas require radian conversion. This calculator accepts all three units and converts automatically — choose what your problem statement uses.
What is the SSA case and why is it called ambiguous?
SSA (two sides and a non-included angle — the angle doesn't lie between the two known sides) is called the "ambiguous case" because the given information can produce zero, one, or two valid triangles. Given side a, side b, and angle A: if a < b·sin A → no triangle exists; if a = b·sin A → exactly one right triangle; if b·sin A < a < b → two different valid triangles; if a ≥ b → exactly one triangle. This calculator evaluates all conditions and returns the valid solution(s), flagging when two distinct triangles are geometrically possible with the same input data.
What is the Law of Sines and when should I use it?
The Law of Sines states: a/sin A = b/sin B = c/sin C — each side is proportional to the sine of its opposite angle. Use it when you know: (1) two angles and any one side (ASA or AAS) — it finds the remaining sides; (2) two sides and a non-included angle (SSA) — the ambiguous case. It's computationally simple but breaks down for the SAS and SSS cases where you don't have an angle opposite to a known side. The Law of Cosines is required for those cases. Most Class 11–12 and JEE triangle problems use a combination of both laws.
How do I find all angles of a triangle if I only know all three sides (SSS)?
With SSS, use the Law of Cosines: cos A = (b² + c² − a²) / (2bc). Rearrange similarly for angle B using sides a and c. Once you have two angles, the third is simply 180° − A − B (all angles in a triangle sum to 180°). Example with the classic 3-4-5 right triangle: cos A = (16 + 25 − 9) / (2 × 4 × 5) = 32/40 = 0.8 → A = arccos(0.8) ≈ 36.87°. Similarly B ≈ 53.13° and C = 90°. This calculator performs all these steps automatically — just enter the three sides under the SSS case.
What are the standard trigonometric values I should memorize for exams?
For Class 10–12 boards and JEE: sin 0°=0, sin 30°=½, sin 45°=1/√2≈0.707, sin 60°=√3/2≈0.866, sin 90°=1. Cos values follow the reverse order: cos 0°=1, cos 30°=√3/2, cos 45°=1/√2, cos 60°=½, cos 90°=0. Tan: 0°=0, 30°=1/√3≈0.577, 45°=1, 60°=√3≈1.732, 90°=undefined. Memory trick for sin: √0/2, √1/2, √2/2, √3/2, √4/2 (i.e., 0, ½, 1/√2, √3/2, 1). For cos, the pattern is reversed. This calculator instantly verifies values for any angle outside these standard ones.