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Degrees: 0–360 • Radians: 0–2π
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Sources & Methodology
All trig functions use JavaScript’s native Math library (IEEE 754 double precision). Right triangle solver applies the Pythagorean theorem and SOH-CAH-TOA. Verified against NCTM mathematics standards.
NCTM — Principles and Standards for School Mathematics
National curriculum reference for trigonometry definitions, right triangle relationships, and unit circle used to verify all formulas and educational content on this page.
Khan Academy — Trigonometry
Reference for SOH-CAH-TOA definitions, unit circle, and inverse function ranges used in the step-by-step explanations and formula boxes.
Trig Functions: Input converted to radians if in degrees (rad = deg × π/180). Math.sin(), Math.cos(), Math.tan() applied. Reciprocals: csc = 1/sin, sec = 1/cos, cot = cos/sin.
Inverse Functions: Math.asin(), Math.acos(), Math.atan() return radians; converted to degrees. Domains: arcsin/arccos require −1 ≤ x ≤ 1; arctan accepts all reals.
Right Triangle: Pythagorean theorem: c² = a² + b². Missing sides from trig ratios. Missing angles from inverse functions. Area = 0.5 × a × b.
Inverse Functions: Math.asin(), Math.acos(), Math.atan() return radians; converted to degrees. Domains: arcsin/arccos require −1 ≤ x ≤ 1; arctan accepts all reals.
Right Triangle: Pythagorean theorem: c² = a² + b². Missing sides from trig ratios. Missing angles from inverse functions. Area = 0.5 × a × b.
⏱ Last reviewed: April 2026
How to Use Trigonometry — Complete Guide
Trigonometry studies the relationship between angles and sides of triangles. The six trigonometric functions — sin, cos, tan, and their reciprocals csc, sec, cot — are the building blocks of geometry, physics, engineering, and calculus. The memory aid SOH-CAH-TOA covers the three primary functions for right triangles.
SOH-CAH-TOA: The Right Triangle Definitions
sin(θ) = Opposite / Hypotenuse (SOH)
cos(θ) = Adjacent / Hypotenuse (CAH)
tan(θ) = Opposite / Adjacent (TOA)
csc(θ) = 1/sin(θ) sec(θ) = 1/cos(θ) cot(θ) = 1/tan(θ)
The hypotenuse is always the side opposite the right angle (the longest side). The opposite and adjacent sides depend on which angle you are working with.
Key Angles Reference Table
| Angle (°) | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 0.5 | 0.866 | 0.577 |
| 45° | π/4 | 0.707 | 0.707 | 1 |
| 60° | π/3 | 0.866 | 0.5 | 1.732 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | 0.866 | −0.5 | −1.732 |
| 135° | 3π/4 | 0.707 | −0.707 | −1 |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Degrees to Radians Conversion
Radians = Degrees × (π / 180)
Degrees = Radians × (180 / π)
Key conversions: 30° = π/6 ≈ 0.524 rad • 45° = π/4 ≈ 0.785 rad
60° = π/3 ≈ 1.047 rad • 90° = π/2 ≈ 1.571 rad • 180° = π ≈ 3.14159 rad
60° = π/3 ≈ 1.047 rad • 90° = π/2 ≈ 1.571 rad • 180° = π ≈ 3.14159 rad
The Pythagorean Identities
- sin²(θ) + cos²(θ) = 1 — fundamental identity from the unit circle
- 1 + tan²(θ) = sec²(θ) — divide the first identity by cos²
- 1 + cot²(θ) = csc²(θ) — divide the first identity by sin²
ASTC Rule — Signs of Trig Functions by Quadrant
The sign (positive or negative) of each trig function depends on which quadrant the angle falls in. The mnemonic All Students Take Calculus (ASTC) gives the pattern:
- Quadrant I (0–90°): All positive
- Quadrant II (90–180°): Sin positive only
- Quadrant III (180–270°): Tan positive only
- Quadrant IV (270–360°): Cos positive only
💡 Calculator Mode Warning: Always verify whether your calculator is in Degree or Radian mode before calculating. This is the most common source of trig errors. sin(90) in degree mode = 1. sin(90) in radian mode = 0.894. This calculator shows both values simultaneously.
Frequently Asked Questions
Sin (sine), cos (cosine), and tan (tangent) are the three primary trigonometric ratios. In a right triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. They extend beyond right triangles through the unit circle definition, which defines them for any angle from 0 to 360 degrees (or 0 to 2π radians).
Multiply degrees by π/180. Example: 90° × (π/180) = π/2 ≈ 1.5708 radians. Key conversions to memorize: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π. To go from radians to degrees, multiply by 180/π.
SOH-CAH-TOA is the standard memory aid for right triangle trig. SOH: Sin = Opposite / Hypotenuse. CAH: Cos = Adjacent / Hypotenuse. TOA: Tan = Opposite / Adjacent. The opposite and adjacent sides are relative to the angle in question. The hypotenuse is always the side opposite the right angle.
Arcsin (sin²), arccos (cos²), and arctan (tan²) reverse the trig functions to find the angle from a ratio. Example: sin(30°) = 0.5, so arcsin(0.5) = 30°. Domains: arcsin and arccos require input between −1 and 1 (the range of sin and cos). Arctan accepts any real number. Results from arcsin and arccos are between 0 and 180°; arctan returns between −90 and 90°.
The unit circle has radius 1 centered at the origin. For any angle measured counterclockwise from the positive x-axis, the x-coordinate of the circle point = cos(angle) and the y-coordinate = sin(angle). This extends trig beyond right triangles to any angle. Tan(angle) = y/x = sin/cos = the slope of the line from origin to the point.
Given any two pieces of information (one angle plus one side, or two sides), you can find everything else. Two sides: use Pythagorean theorem for the third, arctan for the angles. One angle plus hypotenuse: use sin/cos for both legs. One angle plus a leg: use tan for the other leg, sin or cos for the hypotenuse. Use the Right Triangle solver tab above for instant results.
Tan(45°) = 1 exactly. In a 45-45-90 triangle, both legs are equal so opposite/adjacent = 1. Other key values: tan(0°) = 0, tan(30°) = 1/√3 ≈ 0.5774, tan(60°) = √3 ≈ 1.7321, tan(90°) = undefined (the function has a vertical asymptote there), tan(180°) = 0.
These are the reciprocal trig functions. Csc (cosecant) = 1/sin = hypotenuse/opposite. Sec (secant) = 1/cos = hypotenuse/adjacent. Cot (cotangent) = 1/tan = adjacent/opposite = cos/sin. Example: sin(30°) = 0.5, so csc(30°) = 2. These appear frequently in calculus integration and advanced physics.
The fundamental Pythagorean identity is sin²(x) + cos²(x) = 1 for any angle x. This comes directly from the Pythagorean theorem on the unit circle (x² + y² = 1 where x = cos and y = sin). Related: 1 + tan²(x) = sec²(x) and 1 + cot²(x) = csc²(x). Used to simplify trig expressions and prove other identities.
Use inverse trig with the side ratios you know. If you know opposite and hypotenuse: angle = arcsin(opp/hyp). If you know adjacent and hypotenuse: angle = arccos(adj/hyp). If you know opposite and adjacent: angle = arctan(opp/adj). The third angle is always 90° minus the one you found (since the three angles sum to 180° and one is 90°).
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