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Reduction grade 11

Reduction grade 11

Kevinmathscience

21:03

Overview

This video explains the "CAST diagram" (or "All Students Take Calculus" diagram) as a tool to simplify trigonometric expressions. It details how angles are divided into four quadrants, each associated with specific trigonometric functions being positive. The video demonstrates how to use the CAST diagram to reduce angles like 180° ± x and 360° - x to simpler forms, primarily involving x. It also covers special cases for 90° ± x, where trigonometric functions switch to their co-functions (sine to cosine, cosine to sine). Finally, it addresses more complex reductions involving angles greater than 360° or negative angles, showing how to manipulate them into forms recognizable on the CAST diagram.

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Chapters

  • •The CAST diagram is a tool to simplify trigonometry problems.
  • •It divides the coordinate plane into four quadrants.
  • •Angles are measured counter-clockwise from 0° to 360°.
  • •Quadrants are defined by angle ranges: 0-90°, 90-180°, 180-270°, 270-360°.
  • •The diagram helps determine the sign (positive or negative) of trigonometric functions in each quadrant.
  • •Quadrant 1 (0-90°): All functions are positive.
  • •Quadrant 2 (90-180°): Sine is positive.
  • •Quadrant 3 (180-270°): Tangent is positive.
  • •Quadrant 4 (270-360°): Cosine is positive.
  • •The acronym CAST (or All Students Take Calculus) helps remember which function is positive in each quadrant, starting from Quadrant 4 and moving counter-clockwise.
  • •To simplify, locate the angle form (e.g., 360° - x) on the CAST diagram.
  • •The trigonometric function of the reduced angle is the same as the original function (e.g., cos(360° - x) becomes cos(x)).
  • •Determine the sign based on the quadrant where the original angle lies.
  • •For 360° - x (Quadrant 4), cos is positive, so cos(360° - x) = cos(x).
  • •For 180° + x (Quadrant 3), sin is negative, so sin(180° + x) = -sin(x).
  • •These angles involve a switch to the co-function.
  • •sin(90° - x) = cos(x)
  • •cos(90° - x) = sin(x)
  • •sin(90° + x) = cos(x)
  • •cos(90° + x) = -sin(x)
  • •Memorizing these four identities is crucial as they appear frequently.
  • •Angles larger than 360° can be reduced by adding or subtracting multiples of 360°.
  • •Example: cos(540° - x) can be simplified by subtracting 360°.
  • •540° - 360° = 180°, so cos(540° - x) = cos(180° - x).
  • •Then, apply the rules for 180° - x: cos(180° - x) = -cos(x) because cosine is negative in Quadrant 2.
  • •Negative angles can be made positive by adding 360°.
  • •Example: sin(-x) = sin(-x + 360°) = sin(360° - x) = -sin(x).
  • •Angles like x - 180° can be rewritten by adding 360°: x - 180° + 360° = x + 180°.
  • •sin(x + 180°) = -sin(x).
  • •For angles like x - 90°, factor out a negative: sin(x - 90°) = sin(-(90° - x)).
  • •Since sin(-θ) = -sin(θ), this becomes -sin(90° - x).
  • •Using the 90° rule, sin(90° - x) = cos(x), so the final result is -cos(x).

Key Takeaways

  1. 1The CAST diagram is essential for simplifying trigonometric expressions by determining the sign of functions in different quadrants.
  2. 2Angles of the form 180° ± x and 360° - x retain their original trigonometric function but may change sign.
  3. 3Angles of the form 90° ± x change to their co-function (sin to cos, cos to sin) and may also change sign.
  4. 4Angles greater than 360° can be reduced by adding or subtracting multiples of 360°.
  5. 5Negative angles can be made positive by adding 360°.
  6. 6Angles like x - 90° can be manipulated by factoring out a negative sign and applying the rules for negative angles and 90° reductions.
  7. 7Understanding the order of operations and angle manipulation is key to correctly applying these reduction formulas.
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