Trigonometry in Physics

Studying the laws of motion in the 17th century, Newton discovered that the path of a thrown object - a parabola - is described through sine and cosine. Three centuries later, Unity game engine developers implement the same formula in C# for realistic projectile ballistics. Trigonometry isn't an abstraction: it's nature's built-in language, from a pendulum clock to quantum wave functions.

• **Game engines:** R = v²·sin(2θ)/g is computed every time a projectile fires in a shooter or a cannonball launches in a strategy game
• **Audio synthesis:** SHM is the foundation of synthesizers-summing harmonics A·cos(nωt) produces the timbre of any instrument
• **LCD displays:** Malus's law controls the brightness of every pixel through the rotation angle of liquid crystals between crossed polarizers

Предварительные знания

• Graphs of Trigonometric Functions

Projectile Motion and Range

Projectile motion at angle θ splits into two independent components: horizontal (uniform motion) and vertical (free fall). The initial velocity components are computed directly from trigonometry.

**Projectile range formula:** R = v₀²·sin(2θ) / g Maximum range at θ = 45° (sin 90° = 1) Max height: H = v₀²·sin²θ / (2g) Flight time: T = 2v₀·sin θ / g

In game engines, the angle needed to hit a target at (x, y) is found by inverting the range formula: tan θ = (v₀² ± √(v₀⁴ − g(gx² + 2yv₀²))) / (gx). The two roots correspond to a low-arc and a high-arc trajectory - the standard pair of ballistic solutions in game engines.

Simple Harmonic Motion

Simple harmonic motion (SHM) is the fundamental oscillation model in physics. The position x(t) = A·cos(ωt + φ) describes a spring, pendulum, or LC circuit. The parameters match those of a sinusoid: amplitude A, angular frequency ω = 2πf, initial phase φ.

**Simple harmonic motion:** x(t) = A·cos(ωt + φ) v(t) = dx/dt = −Aω·sin(ωt + φ) a(t) = d²x/dt² = −Aω²·cos(ωt + φ) = −ω²·x(t) Velocity leads position by π/2 in phase! Acceleration is always opposite in sign to position.

SHM describes small oscillations of any system near a stable equilibrium. This is why sinusoids are so universal in physics: any potential well near its minimum is approximately parabolic, and motion in a parabolic potential is precisely harmonic.

Wave Superposition and Malus's Law

When two waves of the same frequency superpose, the resultant amplitude depends on their phase difference Δφ. At Δφ = 0 the waves reinforce completely (constructive interference); at Δφ = π they cancel completely (destructive interference).

Malus's law describes the intensity of polarized light after a polarizer: I = I₀·cos²θ, where θ is the angle between the light's polarization direction and the polarizer axis. At θ = 90° the light is completely blocked-the operating principle behind LCD screens and 3D glasses.

**Malus's Law:** I = I₀·cos²θ where I₀ is the incident polarized intensity and θ is the angle between polarizations. θ = 0°: I = I₀ (full transmission) θ = 45°: I = I₀/2 θ = 90°: I = 0 (complete extinction)

Key Ideas

• **Projectile range** R = v₀²·sin(2θ)/g-maximum at θ = 45°; symmetry: θ and (90°−θ) give equal ranges
• **SHM:** x = A·cos(ωt+φ); velocity leads position by π/2-maximum speed at equilibrium, zero at turning points
• **Wave interference:** A_res = 2A·|cos(Δφ/2)|; constructive at Δφ = 2πn, destructive at Δφ = π+2πn
• **Malus's law:** I = I₀·cos²θ-polarized light through a polarizer; the basis of LCDs and 3D cinema

Related Topics

Trigonometry in physics links mathematics to real-world systems:

• Graphs of Trigonometric Functions — SHM is literally a sinusoid in time; the parameters A, ω, φ are the same four parameters seen in graph transformations
• Hyperbolic Functions — Damped oscillations are described by e^(−αt)·cos(ωt)-the product of an exponential and cosine
• Euler's Formula — Writing SHM as Re(A·e^(i(ωt+φ))) turns superposition calculations into complex number arithmetic

Вопросы для размышления

• When launching from a height h, the optimal angle for maximum range is no longer 45°-does it increase or decrease, and why? How does this change aiming strategy in a game?
• Both a spring with stiffness k and a pendulum of length L follow SHM. What plays the role of 'stiffness' for the pendulum? Where do ω = √(k/m) and ω = √(g/L) come from?
• Active noise cancellation creates a sound with Δφ = π relative to ambient noise. Why does this only work well at low frequencies? Consider the accuracy of phase measurement at higher frequencies.

Связанные уроки

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