GELE Mathematics — Plane and Spherical TrigonometryConcept Map

Concept

Trigonometric Functions and Identities

Sub Concepts

• Sine, Cosine, Tangent ratios
• Reciprocal functions (cosecant, secant, cotangent)
• Pythagorean identities
• Double-angle formulas
• Sum and difference formulas
• Unit circle representation

Relationship To Central

Foundation for all trigonometric calculations and manipulations

Concept

Right Triangle Solutions

Sub Concepts

• SOH-CAH-TOA mnemonic
• Complementary angles
• Angle of elevation and depression
• Special angles (30°, 45°, 60°)
• Reference angles

Relationship To Central

Basis for understanding all trigonometric ratios

Concept

Oblique Triangle Solutions

Sub Concepts

• Law of Sines
• Law of Cosines
• Ambiguous case (SSA)
• Triangle area calculations
• Heron's formula

Relationship To Central

Extension to non-right triangles using advanced laws

Concept

Triangle Area Formulas

Sub Concepts

• Area using two sides and included angle
• Heron's semi-perimeter formula
• Area from altitude and base
• Applications in surveying and land measurement

Relationship To Central

Critical for practical engineering applications

Concept

Spherical Trigonometry

Sub Concepts

• Spherical triangles and great circles
• Spherical law of sines
• Spherical law of cosines
• Spherical excess and area
• Navigation and geodetic calculations

Relationship To Central

Extension to 3D curved surfaces for geodesy and astronomy

Concept

Engineering Applications

Sub Concepts

• Surveying and distance measurement
• Structural geometry and force resolution
• Bearing and azimuth calculations
• Vertical angles and heights
• Navigation and triangulation

Relationship To Central

Real-world problems solved using trigonometry

To

Right Triangle Solutions

From

Trigonometric Functions

Strength

strong

Relationship

Functions (sin, cos, tan) are defined from right triangle ratios and enable solving all right triangle problems

To

Oblique Triangle Solutions

From

Right Triangle Solutions

Strength

strong

Relationship

Right triangle concepts form the foundation; oblique triangles are solved by decomposing into right triangles or using laws derived from these ratios

To

Trigonometric Functions

From

Fundamental Identities

Strength

strong

Relationship

Identities are mathematical relationships between trigonometric functions enabling simplification and proof of triangle solutions

To

Oblique Triangle Solutions

From

Law of Sines

Strength

strong

Relationship

Law of Sines is one of two primary methods for solving oblique triangles when angles and opposite sides are known

To

Oblique Triangle Solutions

From

Law of Cosines

Strength

strong

Relationship

Law of Cosines is the second primary method, used when two sides and included angle (SAS) or all three sides (SSS) are known

To

Oblique Triangle Solutions

From

Triangle Area Formulas

Strength

strong

Relationship

Area is calculated after solving the triangle; Heron's formula and ½ab sin C are complementary methods

To

Engineering Applications

From

Oblique Triangle Solutions

Strength

strong

Relationship

Real surveying, structural geometry, and navigation problems require applying triangle solutions systematically

To

Oblique Triangle Solutions

From

Spherical Trigonometry

Strength

moderate

Relationship

Spherical trigonometry extends plane trigonometry to curved surfaces; laws of sines and cosines have spherical analogues

To

Engineering Applications

From

Spherical Trigonometry

Strength

strong

Relationship

Spherical methods are essential for geodesy, large-scale surveying, and navigation on Earth's curved surface

To

Fundamental Identities

From

Trigonometric Functions

Strength

strong

Relationship

Identities are direct manipulations and combinations of the six trigonometric functions

To

Sum and Difference Formulas

From

Double-Angle Formulas

Strength

moderate

Relationship

Double-angle formulas are special cases of sum formulas when both angles are equal (A = B)

To

Trigonometric Functions

From

Unit Circle Representation

Strength

moderate

Relationship

Unit circle geometrically depicts how trigonometric values vary with angle, enabling visualization of all four quadrants

To

Law of Sines

From

Ambiguous Case SSA

Strength

strong

Relationship

Ambiguous case is a critical consideration when using Law of Sines with given side-side-angle configuration; can yield 0, 1, or 2 triangles

To

Right Triangle Solutions

From

Angle of Elevation and Depression

Strength

strong

Relationship

These are practical applications of right triangle trigonometry for measuring heights and distances in surveying

To

Right Triangle Solutions

From

Special Angles

Strength

moderate

Relationship

30°-60°-90° and 45°-45°-90° triangles have memorized ratios enabling rapid calculation without calculators

To

Spherical Trigonometry

From

Spherical Excess

Strength

strong

Relationship

Spherical excess (E = A + B + C - 180°) uniquely characterizes spherical triangles and directly gives area

To

Spherical Trigonometry

From

Great Circles

Strength

strong

Relationship

Great circles are the spherical equivalent of straight lines; sides of spherical triangles are arcs of great circles

To

Engineering Applications

From

Force Resolution

Strength

moderate

Relationship

Trigonometry resolves forces into components using angles between force directions

To

Engineering Applications

From

Bearing and Azimuth Calculations

Strength

strong

Relationship

Navigation and surveying rely on bearing angles and azimuth measurements solved using trigonometry

To

Engineering Applications

From

Triangulation

Strength

strong

Relationship

Surveying uses triangulation networks where trigonometric principles determine unknown distances and positions

To

Heron's Formula

From

Semiperimeter s

Strength

strong

Relationship

Heron's formula requires calculating semiperimeter s = (a + b + c)/2 before computing area

To

Unit Circle Representation

From

Reference Angles

Strength

strong

Relationship

Reference angles simplify calculation of trigonometric values in any quadrant by relating them to first quadrant

To

Right Triangle Solutions

From

Complementary Angles

Strength

moderate

Relationship

In a right triangle, two acute angles are complementary (sum to 90°); sin θ = cos(90° - θ)