GELE Mathematics — Plane and Spherical TrigonometrySummary

For a right triangle with angle θ, the primary functions are defined as: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The reciprocals are csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ. These relationships allow engineers to resolve forces, calculate structural deflections, and determine slopes in embankments and roof pitches. For example, if a retaining wall leans at an angle of 75° from vertical (15° from horizontal), and its height is 6 m, the horizontal projection is 6 tan(15°) ≈ 1.61 m.

The identity sin²θ + cos²θ = 1 is the cornerstone of trigonometric algebra. Dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ. These identities simplify complex expressions and verify trigonometric equations. For instance, simplifying (sin²θ + cos²θ)/(cos²θ) = sec²θ confirms structural stiffness equations under combined loading.

Double-angle formulas are sin(2θ) = 2sinθcosθ, cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1, and tan(2θ) = 2tanθ/(1 − tan²θ). Sum and difference formulas: sin(A ± B) = sinAcosB ± cosAsinB; cos(A ± B) = cosAcosB ∓ sinAsinB; tan(A ± B) = (tanA ± tanB)/(1 ∓ tanAtanB). These are vital when analyzing combined loading, deflections of inclined members, and composite angles in surveying. Example: finding the resultant of two forces at angles 30° and 45° requires angle addition.

For any triangle with sides a, b, c opposite to angles A, B, C respectively: a/sinA = b/sinB = c/sinC. This law applies to oblique triangles and is particularly useful when two angles and one side (AAS or ASA) are known, or when two sides and an angle opposite one of them (SSA — the ambiguous case) are given. Example: In a survey, angles from a baseline to a point are measured as 35° and 60°, with baseline length 100 m. Using the law of sines, the distances to the point can be computed. The law relates linear distances to opposite angles directly, making it essential for triangulation in surveying.

For any triangle: c² = a² + b² − 2ab·cosC (and cyclic permutations). Used when two sides and the included angle (SAS) are known, or when all three sides (SSS) are known and angles are required. Example: A land lot is bounded by two sides of 80 m and 120 m meeting at an angle of 105°; the opposite boundary length is c = √(80² + 120² − 2·80·120·cos105°) = √(6400 + 14400 + 4966) ≈ 160 m. Critical for computing angles in structural trusses, rafter slopes, and lot geometry.

Three primary methods: (1) A = ½·base·height — direct geometric approach; (2) A = ½·a·b·sinC — uses two sides and included angle, convenient when angles are computed; (3) Heron's formula A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2 is the semi-perimeter — used when all three sides are known. Example: For a triangular lot with sides 50, 65, and 75 m: s = 95 m, and A = √[95·45·30·20] = √2,565,000 ≈ 1,601 m². Choice of method depends on available data; Heron's is preferred when sides are known from surveying.

When two sides a, b and angle A (opposite to side a) are given, the law of sines yields sinB = (b·sinA)/a. If sinB > 1, no solution exists. If sinB = 1, one right triangle exists. If sinB < 1 and b > a, one solution exists (angle B is acute). If sinB < 1, b < a, and b > a·sinA, two solutions exist (B could be acute or obtuse); if b ≤ a·sinB, only the acute angle applies. Example: Given a = 30, b = 40, A = 35°: sinB = 40·sin35°/30 ≈ 0.766, so B ≈ 50° or B ≈ 130°. Both produce valid triangles; the surveyor or designer must verify which is physically plausible (e.g., slope angle cannot exceed 90°).

The angle of elevation from point P to point Q is the angle above the horizontal from P's perspective. The angle of depression from Q to P is the same angle below Q's horizontal. If a worker stands 50 m from a building and measures an angle of elevation of 30° to the roof, the roof height is h = 50·tan30° ≈ 28.87 m. Conversely, from the roof, the angle of depression to the worker is 30°. These concepts are fundamental in land surveying, sight-distance calculations on roads per NSCP 2015, and vertical dimension estimation.

A spherical triangle is formed by three arcs on a sphere's surface. Unlike plane triangles, sides and angles are measured in angular units (degrees or radians). The spherical law of sines: sinA/sina = sinB/sinb = sinC/sinc, where a, b, c are the angular side lengths. The spherical law of cosines: cosa = cosb·cosc + sinb·sinc·cosA. The spherical excess E = (A + B + C) − 180° (in degrees) or E = (A + B + C) − π (in radians). The area of a spherical triangle on a sphere of radius R is Area = (πR²·E)/180° (if E in degrees) or Area = R²·E (if E in radians). Example: A spherical triangle on Earth (R ≈ 6,371 km) with angles 100°, 110°, 95° has excess E = 305 − 180 = 125°, and area ≈ (π·6,371²·125)/180 ≈ 6.14 × 10⁷ m².

Bearings measure direction relative to north/south, typically expressed as N30°E (30° east of north) or S45°W (45° west of south). Azimuths measure clockwise from north (0° to 360°). Converting: N30°E = 30° azimuth; S45°W = 225° azimuth. In traverses, successive bearings and distances define a closed loop. For a traverse with known coordinates, forward and back bearings differ by 180°. Trigonometry converts bearing + distance into Cartesian coordinates: if bearing is θ and distance is d, then Δx = d·sinθ, Δy = d·cosθ (using north as y-axis, east as x-axis). This is central to land surveying and establishing construction baselines per RA 544 (Property Registration Decree).

Beyond the core Pythagorean identities, engineers manipulate expressions like (sinθ + cosθ)² = 1 + 2sinθcosθ = 1 + sin(2θ), or tan(A+B) = (tanA + tanB)/(1 − tanAtanB). Verification problems require algebraic steps starting from one side and reaching the other, or reducing both sides to a common form. Example: Prove that (1 − sin²θ)/cos²θ = 1. LHS = cos²θ/cos²θ = 1 = RHS. Such manipulations appear in structural dynamics and vibration analysis where trigonometric functions model periodic behavior.

Two forces F₁ and F₂ at angle θ to each other produce a resultant R = √(F₁² + F₂² + 2F₁F₂·cosθ). This is the law of cosines applied to force triangles. If F₁ = 100 N and F₂ = 150 N at 60°, then R = √(10,000 + 22,500 + 2·100·150·0.5) = √45,000 ≈ 212.1 N. The direction α of the resultant is found using the sine rule: sinα/F₂ = sinθ/R. This concept bridges trigonometry with statics per AISC 360 and reinforced concrete design per ACI 318, appearing in axial + bending load combinations and diagonal brace analysis.

Starting from sin²θ + cos²θ = 1, we derive cos(2θ) = cos²θ − sin²θ. Substituting sin²θ = 1 − cos²θ yields cos(2θ) = 2cos²θ − 1. Similarly, cos(2θ) = 1 − 2sin²θ. These three forms are algebraically equivalent and arise from the same foundational identity.

Both solve oblique triangles but suit different input scenarios. Law of sines is direct when angles are known (AAS, ASA, SSA); law of cosines applies when sides predominate (SAS, SSS). They are complementary tools; a complete triangle solution may alternate between them. For instance, finding side c using law of cosines, then angle A using law of sines.

Heron's formula A = √[s(s−a)(s−b)(s−c)] is derived from the sine formula A = ½ab·sinC by expressing sinC in terms of sides using the law of cosines. It provides an elegant alternative when all three sides are known and height is difficult to determine. The relationship reveals how Heron's encodes the geometric information of all three sides.

Converting between Cartesian coordinates and bearing + distance uses sin and cos addition: if a bearing is composed of two components (e.g., initial bearing plus a deflection), the sum formula applies. Δx = Δ(combined bearing) involves cos(θ₁ + θ₂) or sin(θ₁ + θ₂).

Oblique triangle solutions often decompose the problem into right triangles (altitude method). For instance, dropping a perpendicular from a vertex creates two right triangles, each solved with basic sine/cosine. This bridges the foundational right-triangle functions to more complex scenarios.

Spherical trigonometry extends plane concepts to a curved surface. The law of sines and cosines have analogous forms using angular side lengths. As the sphere radius R → ∞, spherical formulas approximate plane results. Spherical excess E → 0 on a very large sphere, and area formula Area = R²E → the Euclidean formula. This limiting relationship validates both frameworks.

Periodic motion and oscillations in structures (damped vibrations, resonance) are modeled using sinusoidal functions and identities. Resolving forces along principal axes requires angle transformations; modal analysis uses combinations of sine and cosine to decompose responses into harmonic components.

Road design per NSCP 2015 requires sight distance calculations on vertical curves. The angle of elevation from the driver's eye to an obstacle on the road ahead, combined with the curve radius, determines safe speed. Trigonometry links the vertical angle, horizontal distance, and road geometry.