Key Concepts in Trigonometry

1. Similar Triangles

• Definition: Triangles with the same shape but different sizes.
• Properties:
  • Corresponding angles are equal.
  • Ratios of corresponding sides are equal.
• Solving Problems: Use these properties to find missing sides and angles.
• Proving Similarity:
  • Angle-Angle (AA): Two pairs of equal angles.
  • Side-Side-Side (SSS): Ratios of all three pairs of sides are equal.
  • Side-Angle-Side (SAS): Two equal ratios of sides and one equal angle.

2. SOHCAHTOA

• Right Triangle: Triangle with a 90° angle.
• Sides:
  • Hypotenuse (H): Longest side.
  • Opposite (O): Side across from the reference angle.
  • Adjacent (A): Side beside the reference angle.
• Trigonometric Ratios:
  • Sine: Opposite / Hypotenuse
  • Cosine: Adjacent / Hypotenuse
  • Tangent: Opposite / Adjacent
• Applications: Finding missing sides or angles using these ratios.

3. Sine Law and Cosine Law

• Sine Law: Useful when two sides and an angle opposite one of them are known.
• Cosine Law: Useful when two sides and the included angle are known or when all three sides are known.
• Applications: Solving for missing sides or angles in non-right triangles.

4. Special Triangles

• 45°-45°-90° Triangle: Isosceles with legs equal and hypotenuse ( \sqrt{2} ).
  • Ratios: ( \sin 45° = \cos 45° = \frac{1}{\sqrt{2}} ), ( \tan 45° = 1 )
• 30°-60°-90° Triangle: Half of an equilateral triangle.
  • Ratios for 30°: ( \sin 30° = \frac{1}{2}, \cos 30° = \frac{\sqrt{3}}{2}, \tan 30° = \frac{1}{\sqrt{3}} )
  • Ratios for 60°: ( \sin 60° = \frac{\sqrt{3}}{2}, \cos 60° = \frac{1}{2}, \tan 60° = \sqrt{3} )

5. CAST Rule and Unit Circle

• CAST Rule: Helps determine which trigonometric functions are positive in each quadrant.
  • Quadrant I: All positive
  • Quadrant II: Sine positive
  • Quadrant III: Tangent positive
  • Quadrant IV: Cosine positive
• Unit Circle: Circle with radius 1 centered at the origin.
  • Coordinates ((x, y)) give cosine and sine ratios.

6. Exact Values for Angles > 90°

• Using CAST and Reference Angles: Determine sign of trigonometric function based on CAST and find reference angles.

7. Sine and Cosine as Functions

• Graphing: Both are periodic functions.
  • Amplitude: 1 (half the distance between max and min values).
  • Period: 360° for both sine and cosine.

8. Radians

• Definition: Another way to measure angles.
• Conversion:
  • 360° = 2π radians
  • 1° = ( \frac{π}{180} ) radians

9. Trigonometric Identities

• Types:
  • Reciprocal Identities: cosecant, secant, and cotangent.
  • Quotient Identities: tangent and cotangent.
  • Pythagorean Identity: ( \sin^2 x + \cos^2 x = 1 )
• Proofs: Use identities to prove other equations.

10. Solving Trigonometric Equations

• Methodology: Use identities, special triangles, and unit circle to solve equations.
• Principal and General Solutions: Find solutions within a restricted domain and general solutions using periodic properties.