Trigonometric functions — everything you need to know

In a right triangle every acute angle α has three basic ratios. Name the sides relative to α: the opposite side lies across from α, the adjacent side touches α, and the hypotenuse is the longest side across from the right angle.

The three reciprocal functions follow directly: cosecant csc(α) = 1/sin(α), secant sec(α) = 1/cos(α) and cotangent cot(α) = 1/tan(α). Because the ratios only depend on the angle and not on the size of the triangle, similar triangles always give the same value — this is exactly why trigonometry works as a measuring tool.

A useful memory aid: SOH-CAH-TOA — Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

The triangle definition only works for angles between 0° and 90°. The unit circle — a circle of radius 1 centred at the origin — extends the functions to every real number. Take the angle θ measured counter-clockwise from the positive x-axis. The point where the rotating radius meets the circle has the coordinates

So the cosine is the horizontal coordinate and the sine is the vertical coordinate. The tangent is the length of the segment on the vertical line x = 1 cut off by the extended radius, which is why tan θ = sin θ / cos θ blows up whenever cos θ = 0.

Because a full turn is 2π (360°), the point repeats: sine and cosine are periodic with period 2π, tangent and cotangent with period π. Angles are usually given in radians: 180° = π rad, so degrees → radians means multiplying by π/180.

Signs by quadrant (the "All Students Take Calculus" rule): in quadrant I all functions are positive, in II only sine, in III only tangent, in IV only cosine.

Sine: domain all real numbers, range [−1, 1], period 2π, zeros at x = k·π, maxima at π/2 + 2kπ, odd symmetry (point symmetric about the origin: sin(−x) = −sin x).

Cosine: the same curve shifted left by π/2. Domain all real numbers, range [−1, 1], period 2π, zeros at π/2 + kπ, even symmetry (axis symmetric about the y-axis: cos(−x) = cos x).

Tangent: domain all reals except π/2 + kπ, range all real numbers, period π, vertical asymptotes where the cosine vanishes, zeros at k·π, odd symmetry.

The reciprocal functions inherit the poles: secant and cosecant never take values strictly between −1 and 1, and cotangent has asymptotes at k·π.

The general sinusoidal function is

• a — amplitude: stretches the curve vertically; |a| is the distance from the midline to a peak. A negative a mirrors the curve about the x-axis.
• b — angular frequency: compresses horizontally. The period becomes T = 2π / |b| (for tangent: π / |b|).
• c — phase shift: moves the curve right by c (left for negative c). Written as sin(bx + φ) the shift is −φ/b.
• d — vertical shift: raises the midline to y = d, so the range becomes [d − |a|, d + |a|].

Frequency f = 1/T and angular frequency ω = 2πf — that is the physics notation for the same b, used for oscillations y(t) = A·sin(ωt + φ).

For general (non-right) triangles you need the law of sines a/sin α = b/sin β = c/sin γ and the law of cosines c² = a² + b² − 2ab·cos γ, which contains the Pythagorean theorem as the special case γ = 90°.

Because the functions are periodic, an equation has infinitely many solutions. Find one solution with the inverse function (arcsin, arccos, arctan) and then add the period.

The inverse functions themselves are only defined on restricted ranges: arcsin and arctan return values in [−π/2, π/2], arccos in [0, π]. Always check whether the required solutions lie inside the given interval.

With the chain rule, (sin(bx + c))′ = b·cos(bx + c). Differentiating sine four times returns the original function — that cycle is why sine and cosine solve the harmonic oscillator equation y″ = −ω²y.

Useful limit: lim(x→0) sin x / x = 1, which is also the reason the small-angle approximation sin x ≈ x (and cos x ≈ 1 − x²/2) works for small angles in physics.

Example 1 — missing side. A ladder leans against a wall at 65°, its foot 1.8 m away. How long is it? cos(65°) = adjacent/hypotenuse ⇒ hypotenuse = 1.8 / cos(65°) ≈ 4.26 m.

Example 2 — reading a graph. A curve oscillates between y = 1 and y = 7 and repeats every 4 units, starting at its midline going upwards. Then d = (7 + 1)/2 = 4, a = (7 − 1)/2 = 3, b = 2π/4 = π/2, so f(x) = 3·sin(π/2·x) + 4.

Example 3 — an equation. Solve 2·sin x = 1 for x ∈ [0, 2π]. Then sin x = 0.5, so x = π/6 and x = π − π/6 = 5π/6.

Example 4 — a physical model. A mains voltage is U(t) = 325 V · sin(2π·50 Hz · t). Its amplitude is 325 V, its period 1/50 s = 20 ms, and its effective (RMS) value 325/√2 ≈ 230 V.

• Sine and cosine are the same curve, shifted by a quarter period (π/2).
• Every periodic signal can be written as a sum of sines and cosines — that is Fourier analysis, the basis of audio compression, image formats and signal processing.
• Euler's formula ties them to the exponential function: e^(ix) = cos x + i·sin x, and for x = π it gives the famous identity e^(iπ) + 1 = 0.
• Applications: surveying and triangulation, GPS, wave optics, alternating current, pendulums and springs, planetary orbits, computer graphics rotations, and the haversine formula for distances on the globe.
• Historical note: the word "sine" comes from a mistranslation of the Sanskrit jyā ("bowstring") via Arabic into Latin sinus ("bay, fold").