Common Test Mathematics： Formulas to Remember

• WIP

• Things I don’t remember in the IB Math Formula Booklet (by blu3mo)

  • arithmetic $S_n={\sum }_{i=1}^n{u}_i=\frac{n}{2}(u_1+u_n)$
  • geometric $S_n={\sum }_{i=1}^n{u}_1{r}^{i-1}=\frac{u_1(r^n-1)}{r-1}$
    • However, $r\ne 1$
  • polynomial
    • sum $\frac{-a_{n-1}}{a_n}$
    • product $\frac{(-1{)}^n{a}_0}{a_n}$
  • If you’re taking 数2B, you should practice solving polynomial integrals quickly (by takker)
    • Like ${\int }_1^2\left(3x^3+3x+4\right)\mathrm{d}x$
    • When it comes to the exam, being able to solve area calculations using this is quite important
      • I’m not sure about the common tests. I’m sorry.
    • Use the beta function whenever possible
      • Even if you can’t directly apply it, there are cases where you can use the beta function by making a slight transformation
  • triangle $A=\frac{1}{2}ab\sin C$
    • The base of the triangle ($a$) multiplied by the height ($b\sin C$) divided by 2
  • double angle identities
    • $\sin 2\theta =2\sin \theta \cos \theta$
    • $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta =2{\cos }^2\theta -1=1-2{\sin }^2\theta$
  • compound angle identities
    • $\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$
    • $\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$
    • These two identities should definitely be memorized (by takker)
      • There are various ways to remember them, such as using mnemonics or deriving them from $e^{i\theta }=\cos \theta +i\sin \theta$. You can use whichever method works best for you.
    • Whether to memorize other derived identities depends on the person’s opinion
      • It is certain that memorizing them will increase calculation speed
      • However, I think it’s better to be able to derive them easily from these two identities (by takker)
        • It’s much better to be able to quickly derive them than to just memorize the formulas without understanding their derivation
        • In exams at top universities, you are often asked if you understand the principles and derivation behind a formula rather than just whether you have memorized the formula. So being able to derive them on the spot is advantageous, I think.
        • (/icons/Understand)(blu3mo)
        • By the way, the most advantageous case is when you have memorized them while deriving them multiple times (by takker)
          • Practice makes perfect in this case
    • When you are unsure about a formula, it is effective to substitute random values and check
      • For example, if you are not sure whether $\cos 2\theta$ is $2(\cos \theta {)}^2-1$ or $1-2(\cos \theta {)}^2$, you can substitute values like $1$, $0$, or $\frac{1}{2}\pi$ for $\theta$ to confirm
      • In this case,
        • ${\cos \theta |}_{\theta =0}=\cos 0=1$
        • ${2(\cos \theta {)}^2-1|}_{\theta =0}=2\cdot 1^2-1=1$
        • $1-{2(\cos \theta {)}^2|}_{\theta =0}=1-2\cdot 1^2=-1$
      • From this, we can see that $\cos 2\theta =2(\cos \theta {)}^2-1$
      • (/icons/I see)(blu3mo)
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    • Is this necessary for 数2B?
      • Oh, I think trigonometric calculus is in the scope of 数III (by takker)
        • (takker) is not from the era of the common tests, so it’s best to ask your teacher about this
  • The notation of $\mathrm{cosec}\theta ,\sec \theta ,\cot \theta$ has the advantage of omitting fractions, but it can also be confusing in quick calculations (by takker)
    • It’s difficult to calculate $\mathrm{cosec}\theta \sin \theta =1$, but if it’s written as $\frac{1}{\sin \theta }\cdot \sin \theta =1$, you can instantly recognize the form of the equation.
  • You should also be able to derive $\frac{\text{d}}{\text{d}x}\left(\frac{1}{\tan \theta }\right)$
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