Geometry

1. Trigonometry in Geometry
   1. Trigonometric Ratios
      1. Definition and Basics
         1. Sine: ratio of opposite side to hypotenuse in a right triangle
         2. Cosine: ratio of adjacent side to hypotenuse in a right triangle
         3. Tangent: ratio of opposite side to adjacent side in a right triangle
      2. Derived Ratios
         1. Cosecant: reciprocal of sine
         2. Secant: reciprocal of cosine
         3. Cotangent: reciprocal of tangent
      3. Relations between Ratios
         1. Pythagorean identities
            1. sin²θ + cos²θ = 1
            2. 1 + tan²θ = sec²θ
            3. 1 + cot²θ = csc²θ
         2. Sum and difference formulas
         3. Double angle and half angle formulas
   2. Applications in Solving Triangles
      1. Right Triangles
         1. Using trigonometric ratios to find unknown sides
         2. Solving angle measures using inverse trigonometric functions
      2. Oblique Triangles
         1. Using Laws of Sines and Cosines for any triangles
         2. Identifying when to use each law based on given information
      3. Non-Right Triangle Solving Strategies
         1. Reduction to right triangles
         2. Using auxiliary lines
   3. Laws of Sines and Cosines
      1. Law of Sines
         1. a/sinA = b/sinB = c/sinC
         2. Ambiguous case in SSA (Side-Side-Angle) situations
      2. Law of Cosines
         1. c² = a² + b² - 2ab*cosC
         2. Applications for determining unknown side lengths
      3. Applications and Problem Types
         1. Finding area of a triangle using sine
         2. Solving navigation and physics problems with triangles
   4. Trigonometric Functions in Coordinate Geometry
      1. Unit Circle Definition
         1. Relationship between angles and radius
         2. Using the unit circle to understand periodicity and symmetry
      2. Graphical Representations
         1. Sine, cosine, and tangent graphs
         2. Characteristics like amplitude, period, phase shift
      3. Application of Trigonometric Graphs
         1. Modeling waves
         2. Applying to sound and light waves
   5. Advanced Trigonometric Concepts
      1. Polar Coordinates and Trigonometry
         1. Conversion between Cartesian and polar formats
         2. Using polar equations in trigonometric contexts
      2. Complex Numbers and Trigonometry
         1. Euler’s formula: e^(iθ) = cosθ + i*sinθ
         2. Representing complex numbers in polar form
      3. Vector Calculations using Trigonometry
         1. Dot product and angle between vectors
         2. Applying trigonometry to resolve vectors into components