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VCE Specialist Maths formula sheet

Key formulas for VCE Specialist Mathematics: advanced integration, trig identities, vectors and complex numbers.

Integration

  • ∫1a2−x2 dx=sin⁡−1xa+c\displaystyle\int \dfrac{1}{\sqrt{a^2-x^2}}\,dx=\sin^{-1}\dfrac{x}{a}+c
  • ∫1a2+x2 dx=1atan⁡−1xa+c\displaystyle\int \dfrac{1}{a^2+x^2}\,dx=\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+c
  • ∫sec⁡2(ax) dx=1atan⁡(ax)+c\displaystyle\int \sec^2(ax)\,dx=\dfrac{1}{a}\tan(ax)+c

Trig identities

  • sin⁡2x+cos⁡2x=1\sin^2 x+\cos^2 x=1
  • sin⁡2x=2sin⁡xcos⁡x\sin 2x=2\sin x\cos x
  • cos⁡2x=cos⁡2x−sin⁡2x=2cos⁡2x−1=1−2sin⁡2x\cos 2x=\cos^2 x-\sin^2 x=2\cos^2 x-1=1-2\sin^2 x

Vectors

  • Magnitude∣a⃗∣=a12+a22+a32|\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2}
  • Dot producta⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta

Complex numbers

  • z=a+bi,∣z∣=a2+b2z=a+bi,\quad |z|=\sqrt{a^2+b^2}
  • Polar formz=r cis θ=r(cos⁡θ+isin⁡θ)z=r\,\text{cis}\,\theta=r(\cos\theta+i\sin\theta)
  • De Moivrezn=rn cis(nθ)z^n=r^n\,\text{cis}(n\theta)

A study reference compiled by ATAR99. The official VCAA formula/data sheet is provided in your exam — always use that as the authority on the day.

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