Mathematics
"Future Geodetic Engineer! Mathematics - ang foundation ng lahat ng surveying at mapping calculations. Mastering trigonometry, calculus, at statistics para sa accurate measurements!"
1. Plane Trigonometry 📐
Essential for all angle and distance calculations in surveying.
Basic Trig Ratios
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
Mnemonic: SOH-CAH-TOA!
Reciprocal Functions
csc θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
Law of Sines & Cosines
a/sin A = b/sin B = c/sin C (Law of Sines)
c² = a² + b² - 2ab·cos C (Law of Cosines)
Used for: solving oblique triangles in traverse computation, bearing calculations
2. Spherical Trigonometry 🌐
Essential for geodetic surveying - calculating on Earth's curved surface!
| Formula | Application |
|---|---|
| cos a = cos b·cos c + sin b·sin c·cos A | Law of Cosines for Sides |
| cos A = -cos B·cos C + sin B·sin C·cos a | Law of Cosines for Angles |
| sin a/sin A = sin b/sin B = sin c/sin C | Law of Sines (Spherical) |
| Spherical Excess = A + B + C - 180° | Area calculation of spherical triangle |
3. Differential & Integral Calculus 📊
Derivatives (Rate of Change)
- d/dx (xⁿ) = nxⁿ⁻¹ (Power Rule)
- d/dx (sin x) = cos x
- d/dx (cos x) = -sin x
- d/dx (eˣ) = eˣ
- d/dx (ln x) = 1/x
Application: Finding slope of curves, rate of change in leveling
Integration (Area & Volume)
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
- ∫sin x dx = -cos x + C
- ∫cos x dx = sin x + C
- ∫eˣ dx = eˣ + C
Application: Computing areas from cross-sections, earthwork volumes
4. Statistics & Probability 📈
Error analysis and quality control in surveying measurements.
Measures of Central Tendency
- Mean (x̄) = Σx/n
- Median = Middle value
- Mode = Most frequent value
- Weighted Mean = Σ(wx)/Σw
Measures of Dispersion
- Variance (σ²) = Σ(x-x̄)²/n
- Standard Deviation (σ) = √Variance
- Standard Error = σ/√n
- Probable Error = 0.6745σ
Error Propagation in Surveying
For sum/difference: σ_total = √(σ₁² + σ₂² + σ₃²...)
For product: σ_relative = √((σ₁/x₁)² + (σ₂/x₂)²...)
5. Practice Problems 📚
Common Board Exam Problems
Q1: A surveyor measures distances of 150m and 200m at an included angle of 60°. What is the third side?
A: Using Law of Cosines: c² = 150² + 200² - 2(150)(200)cos60° = 22500 + 40000 - 30000 = 32500. c = 180.28m
Q2: The probable error of a single measurement is ±0.02m. What is the probable error of the mean of 16 measurements?
A: PE_mean = PE_single/√n = 0.02/√16 = 0.02/4 = ±0.005m
Q3: Convert 45°30'15" to decimal degrees.
A: 45 + 30/60 + 15/3600 = 45 + 0.5 + 0.00417 = 45.50417°
🔥 GE Challenge 🔥
Master the math behind precise measurements! Trigonometry, calculus, statistics - tools for accurate surveying!
Precision in math = Precision in the field!
Test Your Knowledge! 🧠
Ready ka na ba? Take the practice quiz for Mathematics to reinforce what you just learned.
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