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Next exam on November 14 – 15, 2026:
All concepts

Trigonometric Induction Formulas

三角函数诱导公式

Reduce any angle to an acute one: whether the function name changes depends on the multiple of π/2, and the sign follows the quadrant of the original angle.

Topic drill: learn this step by step, with real exam questions and a mastery check
7 real-exam questions · 5 sittings46 in mock papers7 practice

Formulas

  • Induction Formulas (π ± α, −α)

    Identity
    sin⁡(π±α)=∓sin⁡α,cos⁡(π±α)=−cos⁡αsin⁡(−α)=−sin⁡α,cos⁡(−α)=cos⁡α\begin{aligned}&\sin(\pi\pm\alpha)=\mp\sin\alpha,\quad \cos(\pi\pm\alpha)=-\cos\alpha\\&\sin(-\alpha)=-\sin\alpha,\quad \cos(-\alpha)=\cos\alpha\end{aligned}

  • Induction Formulas (π/2 ± α)

    Identity
    sin⁡(π2±α)=cos⁡α,cos⁡(π2±α)=∓sin⁡α\sin\left(\frac{\pi}{2}\pm\alpha\right)=\cos\alpha,\quad \cos\left(\frac{\pi}{2}\pm\alpha\right)=\mp\sin\alpha

How the real exam tests it

Questions from real CSCA papers that test this concept. Work each one out before you open the answer.

  1. CSCA Math December 2025 Past Paper
    Which trigonometric reduction formula is incorrect?
    • A．tan⁡(π+α)=−tan⁡α\tan(\pi+\alpha)=-\tan\alpha
    • B．cos⁡(2π−α)=cos⁡α\cos(2\pi-\alpha)=\cos\alpha
    • C．sin⁡(π−α)=sin⁡α\sin(\pi-\alpha)=\sin\alpha
    • D．sin⁡(π2−α)=cos⁡α\sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha
    Show answer & solution

    Answer: A

    Answer: A The identity is tan⁡(π+α)=tan⁡α\tan(\pi+\alpha)=\tan\alpha, not −tan⁡α-\tan\alpha. Choose A.
    See it in the full CSCA Math December 2025 Past Paper
  2. CSCA Math January 2026 Past Paper
    Which statement is correct?
    • A．sin⁡(π+α)=sin⁡α\sin ( \pi + \alpha ) = \sin \alpha
    • B．cos⁡(π+α)=cos⁡α\cos ( \pi + \alpha ) = \cos \alpha
    • C．sin⁡(−α)=sin⁡α\sin ( - \alpha ) = \sin \alpha
    • D．cos⁡(−α)=cos⁡α\cos ( - \alpha ) = \cos \alpha
    Show answer & solution

    Answer: D

    Answer: D Cosine is an even function, so cos⁡(−α)=cos⁡α\cos(-\alpha)=\cos\alpha. Therefore D is correct.
    Take the full CSCA Math January 2026 Past Paper — timed & graded
  3. CSCA Math March 2026 Past Paper
    Let α=5π/6\alpha = 5 \pi / 6. Then cos⁡α=\cos \alpha =
    • A．−3/2- \sqrt { 3 } / 2
    • B．3/2\sqrt { 3 } / 2
    • C．−1/2- 1 / 2
    • D．1/21 / 2
    Show answer & solution

    Answer: A

    cos⁡(5π/6)=cos⁡(π−π/6)=−cos⁡(π/6)=−3/2\cos ( 5 \pi / 6 ) = \cos ( \pi - \pi / 6 ) = - \cos ( \pi / 6 ) = - \sqrt { 3 } / 2.
    Take the full CSCA Math March 2026 Past Paper — timed & graded

Real-exam evidence

Recorded past-paper appearances are listed below; online verification is labelled separately.

7 questions across 5 sittings1 of 5 sittings available to verify free online
  • CSCA Math June 2026 Past Paper1 questionreading page not yet public
  • CSCA Math April 2026 Past Paper2 questionsreading page not yet public
  • CSCA Math March 2026 Past Paper2 questionsreading page not yet public
  • CSCA Math January 2026 Past Paper1 questionreading page not yet public
  • CSCA Math December 2025 Past Paper1 questionView question 38 and answer free (full paper)

46 questions across 16 mock papers — our own papers, so a coverage signal rather than real-exam evidence

7 related practice questions availablePractise free: Elementary Function

Sitting evidence comes from the CSCA past papers on record; questions whose original paper is missing are excluded, and sprint papers are deliberately not counted.

Glossary

sine function

Related concepts

Fundamental Trigonometric IdentitiesDouble-Angle FormulasHalf-Angle FormulasTangent Function Graph