Reduction formula

Rules that rewrite angles into the first quadrant. CAPS Grade 12 definition, worked NSC example and related trigonometry practice.

Rules that rewrite angles into the first quadrant.

Definition

Reduction formulas convert trigonometric ratios of angles like 180° − θ, 180° + θ, 360° − θ and −θ into ratios of the acute angle θ, using the CAST diagram to determine the sign.

Worked example

sin(180° − θ) = sinθ, and cos(360° − θ) = cosθ.

Simplify sin(180° + θ)·cos(360° − θ) ÷ tan(−θ) = (−sinθ)(cosθ) ÷ (−tanθ) = sinθ·cosθ × cosθ/sinθ = cos²θ.

Where it appears in the exam

Paper 2 trigonometry asks you to simplify expressions such as sin(180° + θ)·cos(90° − θ) without a calculator.

Step-by-step method

  1. Identify the form of the angle (180° ± θ, 360° − θ, −θ or 90° ± θ).
  2. Use CAST to decide whether the ratio is positive or negative.
  3. For 90° ± θ, switch sin and cos (co-functions).
  4. Simplify the resulting expression.

Common mistake

Getting the sign wrong by skipping the CAST quadrant check.

Related trigonometry terms

Frequently asked questions

What does reduction formula mean in Grade 12 Maths?

Reduction formulas convert trigonometric ratios of angles like 180° − θ, 180° + θ, 360° − θ and −θ into ratios of the acute angle θ, using the CAST diagram to determine the sign.

Which NSC paper tests reduction formula?

Reduction formula belongs to the Trigonometry section of CAPS Grade 12 Mathematics, which is examined in Paper 2.

How do I practise questions involving reduction formula?

Snap a question that uses reduction formula into the Snap&Learn AI solver, or work through the trigonometry topic guide and NSC past papers.

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