Trigonometry: Top 10 Must-Know Topics

1. Similar Triangles

• Definition: Triangles with the same shape but possibly different sizes.
• Properties:
  • Corresponding angles are equal.
  • Ratios of corresponding sides are equal.
• Proving Similarity:
  • Angle-Angle (AA) Similarity: Two pairs of equal angles.
  • Side-Side-Side (SSS) Similarity: Equal ratios of all three pairs of sides.
  • Side-Angle-Side (SAS) Similarity: Two equal side ratios and one equal angle.
• Applications: Solving for unknown sides or angles using corresponding sides and angles.

2. SOHCAHTOA

• Mnemonic for remembering trigonometric ratios:
  • Sine (sin): Opposite / Hypotenuse
  • Cosine (cos): Adjacent / Hypotenuse
  • Tangent (tan): Opposite / Adjacent
• Example Applications:
  • Find missing sides or angles in right triangles.

3. Sine Law and Cosine Law

• Sine Law: Applicable when you know two angles and one side or two sides and a non-included angle.
• Cosine Law: Useful for:
  • Finding a side when two sides and the included angle are known.
  • Finding an angle when all three sides are known.

4. Special Triangles

• 45°-45°-90° Triangle:
  • Sides: 1, 1, √2
  • Ratios: sin 45° = cos 45° = 1/√2, tan 45° = 1
• 30°-60°-90° Triangle:
  • Sides: 1, √3, 2
  • Ratios: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3

5. CAST Rule and Unit Circle

• CAST Rule: Identifies which trigonometric ratios are positive in each quadrant:
  • Quadrant 1: All
  • Quadrant 2: Sine
  • Quadrant 3: Tangent
  • Quadrant 4: Cosine
• Unit Circle:
  • A circle with radius 1 centered at the origin.
  • Coordinates (x, y) represent (cos θ, sin θ).

6. Exact Values of Trigonometric Ratios

• Finding exact values for angles greater than 90° by using reference angles and CAST rule.

7. Sine and Cosine Functions

• Periodic Functions with repeating patterns.
• Amplitude: Half the difference between max and min values (1 for sine and cosine).
• Period: 360° for one complete cycle.

8. Radians

• Alternative angle measurement to degrees.
• Conversion: 360° = 2π radians, 180° = π radians.

9. Trigonometric Identities

• Reciprocal Identities:
  • Cosecant (csc) = 1/sin
  • Secant (sec) = 1/cos
  • Cotangent (cot) = 1/tan
• Quotient Identities:
  • tan x = sin x / cos x
  • cot x = cos x / sin x
• Pythagorean Identity:
  • sin²x + cos²x = 1

10. Solving Trigonometric Equations

• Example:
  • sin x = -1/√2
  • Solutions between 0 and 2π using special triangles and CAST rule.
• Factoring and solving quadratic trigonometric equations.

These notes aim to provide a comprehensive overview of essential trigonometry concepts crucial for solving problems and understanding the subject deeply.