Top 10 Must-Know Concepts in Trigonometry

1. Similar Triangles

• Definition: Triangles with the same shape but possibly different sizes.
• Properties:
  • Corresponding angles are equal (Angle A = Angle D, Angle B = Angle E, Angle C = Angle F).
  • Ratios of corresponding sides are equal (e.g., AB/DE = AC/DF = BC/EF).
• Example: To solve for a length (X) using similar triangles, establish similarity through angle-angle (AA), side-side-side (SSS), or side-angle-side (SAS) methods.

2. SOHCAHTOA

• Acronym to remember the primary trig ratios:
  • Sine = Opposite / Hypotenuse (SOH)
  • Cosine = Adjacent / Hypotenuse (CAH)
  • Tangent = Opposite / Adjacent (TOA)
• Use: Can find missing sides or angles in right triangles.

3. Sine Law and Cosine Law

• Cosine Law: Used for triangles without a right angle. Two versions: one for side length and another for angle.
  • Formula for side: c² = a² + b² - 2ab * cos(C).
  • Rearranged for angle: cosine(C) = (a² + b² - c²) / (2ab).
• Sine Law: Used when you know two sides and the angle opposite one of them. Formula: a/sin(A) = b/sin(B).*

4. Special Triangles

• Isosceles Right Triangle (45°): Ratios for sine, cosine, and tangent:
  • Sine(45°) = 1/√2, Cosine(45°) = 1/√2, Tangent(45°) = 1.
• 30-60-90 Triangle: Ratios derived from halving an equilateral triangle:
  • Sine(30°) = 1/2, Cosine(30°) = √3/2, Tangent(30°) = 1/√3.
  • Sine(60°) = √3/2, Cosine(60°) = 1/2, Tangent(60°) = √3.

5. CAST Rule and the Unit Circle

• Unit Circle: Circle with radius 1 centered at the origin.
  • Sine(θ) = y-coordinate, Cosine(θ) = x-coordinate.
  • Key angles: 0°, 90°, 180°, 270°, 360°.
• CAST Rule: Determines which trig functions are positive in each quadrant:
  • 1st Quadrant: All positive
  • 2nd Quadrant: Sine positive
  • 3rd Quadrant: Tangent positive
  • 4th Quadrant: Cosine positive

6. Finding Exact Values for Angles > 90 Degrees

• Reference Angle: Angle between the terminal arm and the closest x-axis.
• Use the CAST rule to determine the sign of the ratio for angles in different quadrants.

7. Sine and Cosine as Functions

• Graphing sine (y = sin x) and cosine (y = cos x) reveals periodic behavior:
  • Amplitude: 1 (max and min y values are 1 and -1).
  • Period: 360° (one complete cycle).

8. Radians

• Definition: Angle measurement using the arc length divided by the radius.
• Conversion: 180° = π radians.
  • Example: 30° = π/6 radians.

9. Trig Identities

• Fundamental Identities:
  • Reciprocal: [csc(x) = 1/sin(x)] , [sec(x) = 1/cos(x)] , [cot(x) = 1/tan(x)]
  • Quotient: [tan(x) = sin(x)/cos(x)]
  • Pythagorean: [sin^2(x) + cos^2(x) = 1]

10. Solving Trig Equations

• Example: Solve [sin(x) = -1/√2] for x in [0, 2π]. Solutions in quadrants III and IV give:
  • [x_1 = 5π/4, x_2 = 7π/4]
• Example: [2sin^2(x) - 3sin(x) + 1 = 0], factoring yields:
  • Solutions: [x = π/2 + 2πk, x = π/6 + 2πk, x = 5π/6 + 2πk], where k is an integer.