Top 10 Must-Know Concepts in Trigonometry

Jun 4, 2024

Trigonometry: Top 10 Must-Know Concepts

1. Similar Triangles

  • Definition: Triangles with the same shape but different sizes.
  • Properties:
    • Corresponding angles are equal.
    • Ratios of corresponding sides are equal.
  • Solving Example:
    • Prove triangles are similar using angle-angle (AA), side-side-side (SSS), or side-angle-side (SAS) similarity.
    • Use proportions to solve for missing lengths.

2. SOHCAHTOA

  • Acronym for Trig Ratios:
    • Sine (sin): Opposite / Hypotenuse
    • Cosine (cos): Adjacent / Hypotenuse
    • Tangent (tan): Opposite / Adjacent
  • Applications:
    • Find missing side lengths or angles in right triangles.
    • Use calculator for angles not in special triangles.

3. Sine Law and Cosine Law

  • Sine Law:
    • Applies to any triangle: (a/sin A) = (b/sin B) = (c/sin C)
  • Cosine Law:
    • For finding side lengths: c^2 = a^2 + b^2 - 2ab * cos(C)
    • For finding angles: Cos(C) = (a^2 + b^2 - c^2) / (2ab)
  • Usage Scenarios:
    • Sine Law: Two sides and non-included angle, two angles and a side.
    • Cosine Law: Two sides and included angle, three sides.

4. Special Triangles

  • 45°-45°-90° Triangle:
    • Sides in the ratio 1:1:√2.
    • sin(45°) = 1/√2, cos(45°) = 1/√2, tan(45°) = 1.
  • 30°-60°-90° Triangle:
    • Sides in the ratio 1:√3:2.
    • sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3.
    • sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.

5. CAST Rule and Unit Circle

  • Unit Circle:
    • Centered at the origin, radius = 1.
    • Key angles: 0°, 90°, 180°, 270°, and 360°.
    • Cosine equals x-coordinate, sine equals y-coordinate.
  • CAST Rule:
    • Determines positivity of trig ratios in quadrants.
    • All (1st), Sine (2nd), Tangent (3rd), Cosine (4th).
    • Mnemonic: "All Students Take Calculus."

6. Trig Ratios for Angles Beyond 90°

  • Reference Angles:
    • Find reference angle between terminal arm and closest x-axis.
    • Use CAST rule to determine the sign.
  • Example:
    • For 150°: Reference angle = 30°, S(150°) = S(30°) = 1/2.
    • For 225°: Reference angle = 45°, C(225°) = -C(45°) = -√2/2.

7. Sine and Cosine as Functions

  • Sine Function (y = sin(x)):
    • Periodic with a cycle of 360°.
    • Table values at 0°, 90°, 180°, 270°, 360°: (0, 1, 0, -1, 0).
  • Cosine Function (y = cos(x)):
    • Periodic, same cycle length.
    • Table values at same intervals: (1, 0, -1, 0, 1).
  • Properties:
    • Amplitude: 1 (half distance between max and min values).
    • Period: 360°.

8. Radians

  • Definition: Another unit for measuring angles.
  • Conversion:
    • 360° = 2π radians → 1° = π/180 radians.
    • Example: 30° = π/6 radians.

9. Trig Identities

  • Reciprocal Identities:
    • csc(x) = 1/sin(x), sec(x) = 1/cos(x), cot(x) = 1/tan(x).
  • Quotient Identities:
    • tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x).
  • Pythagorean Identity:
    • sin^2(x) + cos^2(x) = 1.
  • Proof Example: Prove tan^2(x) - sin^2(x) = sin^2(x) * tan^2(x).

10. Solving Trig Equations

  • Example 1: Solve sin(x) = -1/√2 on [0, 2π].
    • Reference angle = π/4, Quadrants III and IV.
    • Solutions: 5π/4, 7π/4.
  • Example 2: Solve 2sin^2(x) - 3sin(x) + 1 = 0.
    • Factor: (2sin(x) - 1)(sin(x) - 1) = 0.
    • Solutions: x = π/2 + 2πk, x = π/6 + 2πk, x = 5π/6 + 2πk.

This guide summarizes the key concepts in trigonometry, providing a handy reference for solving problems and understanding the subject better.