Trigonometric Functions Calculator

Understanding the Core Trigonometric Functions

The three primary trigonometric functions—sine, cosine, and tangent—are ratios derived from the sides of a right triangle. In a right triangle with angle α, the hypotenuse is the longest side opposite the 90° angle. The adjacent side forms the angle α and is not the hypotenuse. The opposite side is across from angle α.

These ratios are defined as:

• Sine (sin): the ratio of opposite to hypotenuse
• Cosine (cos): the ratio of adjacent to hypotenuse
• Tangent (tan): the ratio of opposite to adjacent

Trigonometric functions extend beyond acute angles—they apply to all real numbers and angles measured in both degrees and radians. The unit circle provides the geometric foundation for evaluating these functions at any angle.

The Six Trigonometric Functions

The calculator computes three reciprocal pairs. The reciprocal functions are inverses of the primary three:

• α — The angle in question, measured in degrees or radians
• opposite — The side of the triangle opposite to angle α
• adjacent — The side of the triangle adjacent to angle α (not the hypotenuse)
• hypotenuse — The longest side of the right triangle, opposite the right angle

The SOHCAHTOA Memory Aid

A time-tested mnemonic helps recall the definitions of the three primary functions:

• SOH: Sine = Opposite ÷ Hypotenuse
• CAH: Cosine = Adjacent ÷ Hypotenuse
• TOA: Tangent = Opposite ÷ Adjacent

This phrase—phonetically sounding like "soak-a-toe"—has helped generations of students lock in these ratios without confusion. Once you know the reciprocal relationships (cosecant is 1/sine, secant is 1/cosine, cotangent is 1/tangent), the full set of six functions becomes straightforward to recall and apply.

Degrees Versus Radians

Angles can be measured in degrees (where a full rotation equals 360°) or radians (where a full rotation equals 2π). The conversion formula is:

• Angle (radians) = Angle (degrees) × π ÷ 180
• Angle (degrees) = Angle (radians) × 180 ÷ π

When entering an angle like π/3, select the "π radians" unit and input 1/3. The calculator automatically converts between unit systems and evaluates all six functions in the angle system you choose. Most calculus and advanced mathematics use radians, while engineering and navigation often prefer degrees.

Common Pitfalls and Considerations

Avoid these frequent mistakes when working with trigonometric functions:

1. Tangent and cotangent are undefined at certain angles — Tangent is undefined when cos(α) = 0, which occurs at 90°, 270°, and other odd multiples of 90°. Cotangent is undefined when sin(α) = 0, happening at 0°, 180°, 360°, and their multiples. The calculator will flag these cases rather than return a false value.
2. Reciprocal functions have restricted ranges — Cosecant and secant must satisfy |csc(α)| ≥ 1 and |sec(α)| ≥ 1. Unlike sine and cosine (which range from −1 to 1), their reciprocals cannot exist between −1 and 1. This constraint eliminates many potential angle values for certain applications.
3. Angle measurement consistency matters — Switching between degrees and radians mid-calculation is a frequent source of error. π/6 radians equals 30°, not 6°. Keep your unit system consistent throughout, and use this calculator to verify conversions when uncertain.
4. Periodic behavior repeats every 360° or 2π radians — Trigonometric functions are periodic. sin(45°) = sin(405°) = sin(765°). When working with angles beyond one full rotation, reduce them to their equivalent within 0°–360° (or 0–2π) to avoid confusion and verify your inputs match your expected range.