Plane and Spherical Trigonometry - Concept Map
Trigonometric Functions and Triangle Solutions
Related concepts
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
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
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
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
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
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
Concept connections
- Trigonometric FunctionsRight Triangle SolutionsFunctions (sin, cos, tan) are defined from right triangle ratios and enable solving all right triangle problems
- Right Triangle SolutionsOblique Triangle SolutionsRight triangle concepts form the foundation; oblique triangles are solved by decomposing into right triangles or using laws derived from these ratios
- Fundamental IdentitiesTrigonometric FunctionsIdentities are mathematical relationships between trigonometric functions enabling simplification and proof of triangle solutions
- Law of SinesOblique Triangle SolutionsLaw of Sines is one of two primary methods for solving oblique triangles when angles and opposite sides are known
- Law of CosinesOblique Triangle SolutionsLaw of Cosines is the second primary method, used when two sides and included angle (SAS) or all three sides (SSS) are known
- Triangle Area FormulasOblique Triangle SolutionsArea is calculated after solving the triangle; Heron's formula and ½ab sin C are complementary methods
- Oblique Triangle SolutionsEngineering ApplicationsReal surveying, structural geometry, and navigation problems require applying triangle solutions systematically
- Spherical TrigonometryOblique Triangle SolutionsSpherical trigonometry extends plane trigonometry to curved surfaces; laws of sines and cosines have spherical analogues
- Spherical TrigonometryEngineering ApplicationsSpherical methods are essential for geodesy, large-scale surveying, and navigation on Earth's curved surface
- Trigonometric FunctionsFundamental IdentitiesIdentities are direct manipulations and combinations of the six trigonometric functions
- Double-Angle FormulasSum and Difference FormulasDouble-angle formulas are special cases of sum formulas when both angles are equal (A = B)
- Unit Circle RepresentationTrigonometric FunctionsUnit circle geometrically depicts how trigonometric values vary with angle, enabling visualization of all four quadrants
- Ambiguous Case SSALaw of SinesAmbiguous case is a critical consideration when using Law of Sines with given side-side-angle configuration; can yield 0, 1, or 2 triangles
- Angle of Elevation and DepressionRight Triangle SolutionsThese are practical applications of right triangle trigonometry for measuring heights and distances in surveying
- Special AnglesRight Triangle Solutions30°-60°-90° and 45°-45°-90° triangles have memorized ratios enabling rapid calculation without calculators
- Spherical ExcessSpherical TrigonometrySpherical excess (E = A + B + C - 180°) uniquely characterizes spherical triangles and directly gives area
- Great CirclesSpherical TrigonometryGreat circles are the spherical equivalent of straight lines; sides of spherical triangles are arcs of great circles
- Force ResolutionEngineering ApplicationsTrigonometry resolves forces into components using angles between force directions
- Bearing and Azimuth CalculationsEngineering ApplicationsNavigation and surveying rely on bearing angles and azimuth measurements solved using trigonometry
- TriangulationEngineering ApplicationsSurveying uses triangulation networks where trigonometric principles determine unknown distances and positions
- Semiperimeter sHeron's FormulaHeron's formula requires calculating semiperimeter s = (a + b + c)/2 before computing area
- Reference AnglesUnit Circle RepresentationReference angles simplify calculation of trigonometric values in any quadrant by relating them to first quadrant
- Complementary AnglesRight Triangle SolutionsIn a right triangle, two acute angles are complementary (sum to 90°); sin θ = cos(90° - θ)
Diagrams
Complete Concept Hierarchy – Plane and Spherical Trigonometry
Comprehensive mind map showing the hierarchical structure of plane and spherical trigonometry, organized from fundamental functions through identities and into specialized applications. This map displays how right triangle solutions form the foundation, leading to oblique triangle solutions via the law of sines and cosines, and extending to spherical geometry for three-dimensional problems.
Triangle Solution Decision Tree
Decision flowchart for solving triangles systematically. Shows the logical path from identifying given information through to complete triangle solution, including the critical ambiguous case (SSA) and area calculation methods. Essential for board-exam problem-solving strategy.
Oblique Triangle Solution Step-by-Step Process
Step-by-step solution procedure for oblique triangles, showing the logical sequence from data identification through to final verification. Demonstrates both SAS (Law of Cosines first) and SSS (all angles from Law of Cosines) cases, essential for licensure exam problem-solving.
Trigonometric Identity Application Strategy
Strategic approach to selecting and applying trigonometric identities based on expression characteristics. Guides students through identity selection and application for simplification and proof problems common in licensure exams.
Practical Engineering Problem-Solving Sequence
Complete problem-solving sequence for real-world surveying and structural geometry problems. Essential framework for board exam success, integrating problem reading, sketching, data analysis, formula selection, and verification—critical for engineering licensure.
Spherical Trigonometry Application for Geodesy
Specialized workflow for applying spherical trigonometry to geodetic and navigation problems. Demonstrates how plane trigonometry extends to Earth-based surveying and shows the critical role of spherical excess in area calculations—important for advanced surveying topics on licensure exams.
Distribution of Trigonometry Topics in Licensure Exams
Statistical distribution of trigonometry topics appearing in PRC Civil Engineer licensure examinations. Shows that right triangle applications dominate (~25%), followed closely by oblique triangle solutions via Law of Sines and Cosines (~40% combined), with identity proofs and specialized topics making up the remainder. Guides study priority and time allocation.