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Reduction grade 11: Using Numbers and special angles.

Reduction grade 11: Using Numbers and special angles.

Kevinmathscience

16:49

Overview

This video explains how to reduce trigonometric functions involving specific angles, building upon the previous lesson on the CAST diagram. It demonstrates how to simplify expressions like tan(200°) by rewriting the angle in terms of reference angles found on the CAST diagram (e.g., 180° + 20°). The lesson then introduces the concept of 'special triangles' (30-60-90 and 45-45-90) as a method to find exact trigonometric values for common angles without a calculator. Finally, it explains the complementary angle identity, where sin(x) = cos(90° - x), and how this can be used to simplify expressions, particularly in fractions.

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Chapters

  • •Review of reducing trigonometric functions using the CAST diagram with expressions like tan(360° - x).
  • •Method for handling angles not directly on the CAST diagram by subtracting multiples of 360°.
  • •Transition from reducing expressions with variables (like x) to reducing specific numerical angles.
  • •Locating the numerical angle on the CAST diagram.
  • •Rewriting the angle to match a form on the CAST diagram (e.g., 200° as 180° + 20°).
  • •Applying the trigonometric function and sign based on the quadrant.
  • •Reducing sin(200°) to -sin(20°).
  • •Reducing sin(340°) to -sin(20°).
  • •Reducing tan(300°) to -tan(60°).
  • •Reducing sin(150°) to sin(30°).
  • •Reducing tan(730°) by repeatedly subtracting 360° to get tan(10°).
  • •Reducing sin(500°) by subtracting 360° to get sin(140°), then reducing to sin(40°).
  • •Reducing tan(-100°) by adding 360° to get tan(260°), then reducing to tan(80°).
  • •Explanation of why special triangles (30-60-90 and 45-45-90) are used.
  • •Memorizing these triangles allows for exact calculations without a calculator.
  • •These triangles are useful because the resulting angles after reduction are often common angles like 30°, 45°, or 60°.
  • •Example: Reducing sin(300°) to -sin(60°) and using the 60° special triangle to find the exact value (-√3/2).
  • •Example: Reducing cos(-60°) to cos(300°), then to cos(60°), and using the triangle (1/2).
  • •Example: Reducing tan(210°) to tan(30°) and using the triangle (1/√3).
  • •Example: Reducing cos(390°) to cos(30°) and using the triangle (√3/2).
  • •The identity: sin(x) = cos(90° - x) and cos(x) = sin(90° - x).
  • •Examples: sin(20°) = cos(70°), cos(10°) = sin(80°).
  • •Application in simplifying fractions where complementary angles can cancel out.

Key Takeaways

  1. 1Angles outside the 0°-360° range can be simplified by adding or subtracting multiples of 360°.
  2. 2Any angle can be rewritten in terms of a reference angle on the CAST diagram (e.g., 180° ± x, 360° ± x).
  3. 3The sign of the trigonometric function depends on the quadrant the original angle falls into.
  4. 4Special triangles (30-60-90 and 45-45-90) provide exact values for common trigonometric angles.
  5. 5Memorizing special triangles is crucial for avoiding calculator use and showing work in exams.
  6. 6The complementary angle identity (sin(x) = cos(90°-x)) allows for simplification, especially in fractions.
  7. 7Combining CAST diagram rules with special triangles and complementary identities enables full reduction of trigonometric expressions.
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Reduction grade 11: Using Numbers and special angles. | NoteTube | NoteTube