Category Archives: Precalculus

Some Important Formulas from Algebra and Trigonometry

I think it would be a good idea to review some important formulas from algebra and trigonometry before we get into serious stuff in calculus.

Expansion of Polynomials

  1. \((a+b)^2=a^2+2ab+b^2\)
  2. \((a-b)^2=a^2-2ab+b^2\)
  3. \((a+b)^3=a^3+3a^2b+3ab^2+b^3\)
  4. \((a-b)^3=a^3-3a^2b+3ab^2-b^3\)

Factorization of Polynomials

  1. \(a^2-b^2=(a+b)(a-b)\)
  2. \(a^3-b^3=(a-b)(a^2+ab+b^2)\)
  3. \(a^3+b^3=(a+b)(a^2-ab+b^2)\)

Trigonometric Identities

  1. \(\cos^2\theta+\sin^2\theta=1\)
  2. \(\tan^2\theta+1=\sec^2\theta\)

Sine Sum and Difference Formulas

  1. \(\sin(\theta_1+\theta_2)=\sin\theta_1\cos\theta_2+\cos\theta_1\sin\theta_2\)
  2. \(\sin(\theta_1-\theta_2)=\sin\theta_1\cos\theta_2-\cos\theta_1\sin\theta_2\)

Sine Double Angle Formula \[\sin2\theta=2\sin\theta\cos\theta\]

Cosine Sum and Difference Formulas

  1. \(\cos(\theta_1+\theta_2)=\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2\)
  2. \(\cos(\theta_1-\theta_2)=\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\)

Cosine Double Angle Formula \begin{eqnarray*}\cos2\theta&=&\cos^2\theta-\sin^2\theta\\&=&2\cos^2\theta-1\\&=&1-2\sin^2\theta\end{eqnarray*}

From this Cosine Double Angle Formula, we obtain Half Angle Formulas.

Half Angle Formulas

  1. \(\cos^2\theta=\displaystyle\frac{1+\cos\theta}{2}\) or equivalently \(\cos\theta=\pm\sqrt{\displaystyle\frac{1+\cos\theta}{2}}\)
  2. \(\sin^2\theta=\displaystyle\frac{1-\cos\theta}{2}\) or equivalently \(\sin\theta=\pm\sqrt{\displaystyle\frac{1-\cos\theta}{2}}\)