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Find all angles from 0° to 360° with a given ratio

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Useful for: MCR3U IB Math AA SL & HL AP® Precalculus

Learn it Find the reference angle from the absolute value of the ratio, then use CAST to decide which two quadrants fit the sign, and write both angles.

Example: Solve sin θ = −0.6 for 0° ≤ θ ≤ 360°.

  1. Reference angle: sin⁻¹(0.6) ≈ 36.9°.
  2. Sine is negative in Q3 and Q4: θ ≈ 180° + 36.9° = 216.9° and 360° − 36.9° = 323.1°.

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