Core Trigonometry Concepts and Techniques

Overview

• Lecture reviews core trigonometry concepts and problem-solving techniques.
• Topics include degree-radian conversion, coterminal angles, arc length, trig ratios, identities, reference angles, quadrants, special triangles, and exact values.
• Emphasis on using algebraic identities and reference triangles instead of calculators.

Degree and Radian Conversion

• Convert degrees to radians: multiply by pi/180.
• Convert radians to degrees: multiply by 180/pi.
• Example: 60° = 60·π/180 = π/3.
• Example: –5π/6 = (–5·180)/6 = –150°.

Coterminal Angles

• Coterminal angles differ by full rotations: add or subtract 360° (or 2π radians).
• For radians: add/subtract 2π. Example: 5π/8 + 2π = 21π/8; subtracting gives –11π/8.

Arc Length

• Formula: s = r·θ where θ is in radians.
• Convert angle to radians first.
• Example: θ = 120° = 2π/3; r = 9 in → s = 9·(2π/3) = 6π ≈ 18.85 in.

Special Right Triangles And Pythagorean Triples

• Common triples: (3,4,5), (5,12,13), (7,24,25), (8,15,17), (9,40,41), (11,60,61).
• Multiples of triples also valid (e.g., 6,8,10).
• Use a^2 + b^2 = c^2 to find missing side.

SOHCAHTOA (Basic Trig Ratios)

• Sine: sin θ = opposite / hypotenuse.
• Cosine: cos θ = adjacent / hypotenuse.
• Tangent: tan θ = opposite / adjacent.
• Use triangle labeling to assign opposite/adjacent relative to angle.

Reciprocal Identities

• csc θ = 1 / sin θ.
• sec θ = 1 / cos θ.
• cot θ = 1 / tan θ.
• Relationship: tan θ = sin θ / cos θ.

Solving Ratio Problems (Examples)

• Given a 3-4-5 triangle and angle x with opposite=4, hypotenuse=5 → sin x = 4/5.
• Given a 5-12-13 triangle with adjacent=12, hypotenuse=13 → sec x = 13/12.
• Given sin x = 7/25 in quadrant I (7-24-25 triangle) → tan x = 7/24.
• Given tan x = –8/15 with x in Q4 (8-15-17 triangle) → sin x = –8/17 so csc x = –17/8.

Pythagorean Identities

• sin^2 θ + cos^2 θ = 1.
• 1 + cot^2 θ = csc^2 θ.
• 1 + tan^2 θ = sec^2 θ.
• Use these to compute one ratio from another. Example: if sin x = 5/7 and x in QII → cos x = –(2√6)/7.

Quadrants And Sign Rules

• Quadrant I: sin +, cos +, tan +.
• Quadrant II: sin +, cos –, tan –.
• Quadrant III: sin –, cos –, tan +.
• Quadrant IV: sin –, cos +, tan –.
• Mnemonic: "All Students Take Calculus" → All (QI), Sine (QII), Tangent (QIII), Cosine (QIV).

Even/Odd Trig Functions

• Sine, tangent, cosecant, cotangent are odd: f(–x) = –f(x).
• Cosine and secant are even: f(–x) = f(x).
• Example: sin(–π/3) = –sin(π/3).

Cofunction Identities

• sin θ = cos(π/2 – θ) or in degrees sin θ = cos(90° – θ).
• cos θ = sin(π/2 – θ).
• tan θ = cot(π/2 – θ), cot θ = tan(π/2 – θ).
• sec θ = csc(π/2 – θ), csc θ = sec(π/2 – θ).
• Example: sin(π/5) = cos(π/2 – π/5) = cos(3π/10).

Reference Angles

• Definition: acute angle between terminal side and x-axis (0°–90° range).
• Compute by quadrant:
  • QI: reference = θ.
  • QII: reference = 180° – θ.
  • QIII: reference = θ – 180°.
  • QIV: reference = 360° – θ.
• Example: reference angle of 290° = 360° – 290° = 70°.

Exact Values Using Reference Triangles and Unit Circle

• 30°–60°–90° triangle: sides 1 (opposite 30°), √3 (opposite 60°), 2 (hypotenuse).
  • cos 60° = 1/2, tan 30° = (1)/(√3) = √3/3.
• 45°–45°–90° triangle: legs 1, hypotenuse √2.
  • sin 45° = √2/2.
• Unit circle: cos θ = x-coordinate, sin θ = y-coordinate.
• Apply quadrant sign to exact values.
• Example: cos 210°: reference 30°, cos 30° = √3/2, QIII → cos 210° = –√3/2.

Example Problems Summary Table

Key Terms And Definitions

• Terminal Side: ray where angle ends; initial side: where angle starts.
• Coterminal Angles: angles differing by full rotations (360° or 2π).
• Reference Angle: acute angle between terminal side and x-axis.
• Special Triangles: memorized triangles used for exact values.
• Even/Odd Functions: parity properties for trig functions.

Action Items / Next Steps

• Memorize special right triangles (30–60–90 and 45–45–90) and common triples.
• Practice converting degrees↔radians and applying sign rules by quadrant.
• Practice using identities (Pythagorean, reciprocal, cofunction, even/odd) without a calculator.
• When allowed, verify exact answers with a calculator in correct mode (degree vs radian).