Trigonometry

Trigonometry links angles and lengths and describes all oscillations. Learn degrees and radians, the unit circle, sine, cosine, tangent and reciprocal functions, inverse trig functions, key identities (Pythagorean, compound, double and half angle, product-to-sum), solving trig equations, the sine and cosine rules, and the amplitude–phase form used in AC circuits and vibrations.

From surveying a building's height to analysing a 50 Hz AC supply or a robot joint, trigonometry is indispensable. The functions sin and cos are the natural language of rotation and oscillation.

An analogy

Picture a point moving around a circle: its height traces a sine wave and its horizontal position traces a cosine wave — that's why rotating machines produce sinusoidal signals.

Right triangles and the unit circle

SOH-CAH-TOA for right triangles; the unit circle extends sin and cos to any angle with signs by quadrant (All-Sin-Tan-Cos). Calculators must be in the right mode — degree/radian confusion is a classic error.

Identities and equations

Identities simplify expressions and integrals. Trig equations have infinitely many solutions: sin x = 0.5 → x = 30° + 360°k or 150° + 360°k. Inverse functions return principal values only.

Non-right triangles

Sine rule when you know an angle–side pair; cosine rule for two sides and the included angle (or three sides). Used in surveying, truss geometry and navigation.

Sinusoids

y = A sin(ωt + φ): amplitude A, frequency f = ω/2π, phase φ. Sums of same-frequency sinusoids combine into one sinusoid (phasor addition). RMS of a sine is A/√2 (325 V peak → 230 V RMS).

Formulas

• Radians: π rad = 180°
• Pythagorean identity: sin²θ + cos²θ = 1
• Compound angle: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB
• Double angle: sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ
• Sine & cosine rules: a/sinA = b/sinB = c/sinC; c² = a² + b² − 2ab cos C
• Amplitude–phase: a cos x + b sin x = R cos(x − α), R = √(a² + b²), α = atan2(b, a)

Worked example: Sine and cosine rules

Given: Triangle: a = 10, A = 30°, B = 45°; Another: a = 7, b = 9, C = 60°

Find: b and c.

• b = 10 sin45°/sin30° = 14.14
• c² = 49 + 81 − 2(7)(9)(0.5) = 67 → c = 8.19

Answer: b = 14.14, c = 8.19

Worked example: AC waveform

Given: v = 325 sin(314t) V

Find: Peak, RMS, frequency.

• Peak 325 V
• RMS = 325/√2 = 229.8 V
• f = 314/2π = 50 Hz

Answer: 230 V RMS, 50 Hz

Common mistakes

• Wrong: Calculator in degrees for calculus/radian formulas. — Right: ω t is in radians; switch modes consciously.
• Wrong: Taking only the principal inverse-trig solution. — Right: Add all solutions in the required interval.

Applications

• AC circuits: Phasors, power factor, three-phase systems.
• Surveying & navigation: Heights, distances, bearings.
• Vibration & robotics: Harmonic motion and joint kinematics.

Key takeaways

• π rad = 180°.
• Key identities: Pythagorean, compound, double angle.
• Sine/cosine rules for any triangle.
• A sin(ωt + φ): amplitude, frequency, phase.

Going further

• Hyperbolic functions: sinh, cosh in cables and transmission lines.
• Euler's formula: e^(jθ) = cosθ + j sinθ — see Complex Numbers.
• Spherical trigonometry: Navigation on Earth's surface.