# 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. $\cos^2\theta=\displaystyle\frac{1+\cos2\theta}{2}$ or equivalently $\cos\theta=\pm\sqrt{\displaystyle\frac{1+\cos2\theta}{2}}$
2. $\sin^2\theta=\displaystyle\frac{1-\cos2\theta}{2}$ or equivalently $\sin\theta=\pm\sqrt{\displaystyle\frac{1-\cos2\theta}{2}}$