Trigonometry: Top 10 Must-Know Concepts

1. Similar Triangles

• Similar Triangles: Same shape, different sizes.
• Properties:
  • Corresponding angles are equal.
  • Ratios of corresponding sides are equal.
• Proof of Similarity:
  • Angle-Angle (AA) similarity.
  • Side-Side-Side (SSS) similarity.
  • Side-Angle-Side (SAS) similarity.
• Example: Using properties to solve for missing sides/angles.

2. SOHCAHTOA

• Acronym: Helps remember primary trig ratios.
• Right Triangle:
  • Hypotenuse (H)
  • Opposite (O)
  • Adjacent (A)
• Ratios:
  • Sine (O/H)
  • Cosine (A/H)
  • Tangent (O/A)
• Applications: Solve for missing sides or angles using trig functions.

3. Sine Law and Cosine Law

• Non-right triangles: Use these laws to solve missing sides/angles.
• Cosine Law:
  • Use when two sides and included angle are known.
  • Rearranged for angle when all sides are known.
• Sine Law:
  • Use when two angles and a side, or two sides and a non-included angle are known.

4. Special Triangles

• Isosceles Right Triangle:
  • Angles: 45°, 45°, 90°
  • Ratios: 1:1:√2
• Half Equilateral Triangle:
  • Angles: 30°, 60°, 90°
  • Ratios: 1:√3:2
• Use: Find exact values for sine, cosine, and tangent of special angles.

5. CAST Rule and Unit Circle

• Unit Circle:
  • Circle with radius 1, centered at origin.
  • Coordinates give cosine (x) and sine (y) values.
• CAST Rule:
  • Helps remember which trig functions are positive in which quadrant.
  • Quadrants: All Students Take Calculus (All, Sine, Tangent, Cosine).

6. Finding Exact Trig Values

• Use special triangles and CAST rule to find the exact value of trig functions for any angle.

7. Sine and Cosine as Functions

• Periodic Functions:
  • Sine and cosine oscillate with specific amplitude and period.
• Graphs:
  • Sine: Starts at 0, oscillates between -1 and 1.
  • Cosine: Starts at 1, oscillates similarly.

8. Radians

• Definition: Measure of angle based on arc length/radius.
• Conversion: 360° = 2π radians.
• Use: Convert degrees to radians using π/180.

9. Trig Identities

• Reciprocal Identities:
  • Cosecant, Secant, Cotangent are reciprocals of Sine, Cosine, Tangent.
• Quotient Identities:
  • Tangent = Sine/Cosine; Cotangent = Cosine/Sine.
• Pythagorean Identity:
  • sin²x + cos²x = 1.

10. Solving Trig Equations

• Types of Equations: Range-limited and unrestricted.

• Methods:

  • Use special triangles, CAST rule, and identities.
  • Quadratic-like equations can be factored for solutions.

• Example Problem: Solve within a specific interval or overall.