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Core Trigonometry Concepts

Sep 9, 2025

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Overview

This lecture covers core trigonometry concepts, including angle conversions, reference angles, coterminal angles, right triangle ratios, trig identities, and how to solve common trig problems without a calculator.

Angle Conversion: Degrees and Radians

  • To convert degrees to radians, multiply degrees by π/180.
  • To convert radians to degrees, multiply radians by 180/π.
  • Example: 60° = π/3 radians.

Coterminal Angles

  • Coterminal angles share the same terminal side; found by adding or subtracting 360° (or 2π radians).
  • Example: 5π/8 + 2π = 21π/8 is coterminal with 5π/8.

Arc Length

  • Arc length formula: Arc = angle (in radians) × radius.
  • Convert angle to radians before using the formula.

Right Triangle Trigonometry

  • SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
  • Special triangles include 3-4-5, 5-12-13, 7-24-25, 8-15-17.

Reciprocal and Pythagorean Identities

  • sec x = 1/cos x; csc x = 1/sin x; cot x = 1/tan x.
  • sin²x + cos²x = 1; 1 + tan²x = sec²x; 1 + cot²x = csc²x.

Trigonometric Function Signs by Quadrant

  • Quadrant I: All positive; II: sin positive; III: tan positive; IV: cos positive.
  • "All Students Take Calculus" helps remember the signs.

Even and Odd Trig Functions

  • sin(-x) = -sin(x); cos(-x) = cos(x).
  • tan, cot, csc are odd functions; sec is even.

Cofunction Identities

  • sin θ = cos(90°-θ) or cos(π/2-θ).
  • tan θ = cot(90°-θ).

Reference Angles

  • Quadrant II: 180° - angle; III: angle - 180°; IV: 360° - angle.
  • Reference angle is always acute (0-90°).

Unit Circle and Exact Values

  • Know common reference triangles: 30-60-90 and 45-45-90.
  • Cos 60° = 1/2; sin 45° = √2/2; tan 30° = √3/3.
  • Use x = cos θ, y = sin θ in unit circle.

Solving Trig Problems Without Calculator

  • Use identities, triangles, or unit circle to solve for exact values.
  • Rationalize denominators when needed.

Key Terms & Definitions

  • Coterminal Angles — Angles differing by full rotations (360° or 2π).
  • Reference Angle — The smallest positive angle between the terminal side and the x-axis.
  • Reciprocal Identities — Identities expressing trig functions as reciprocals (e.g., sec x = 1/cos x).
  • Pythagorean Identities — Fundamental identities like sin²x + cos²x = 1.
  • Cofunction Identities — Pairing of sine/cosine, tangent/cotangent, secant/cosecant based on complementary angles.

Action Items / Next Steps

  • Practice converting between degrees and radians.
  • Memorize special right triangles and their side ratios.
  • Learn and use key trigonometric identities.
  • Review the signs of trig functions in all four quadrants.
  • Complete any assigned homework or practice problems on angle conversions and reference angles.

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