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Induction in Sage

Here is an example of proving something using induction in Sage. Suppose we want to prove that: \begin{equation*}...

Posted by
Beetle B.
on
Wed 24 November 2021

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Areas of Triangles

When You Know 2 Sides and 1 Angle \begin{equation*} A=\frac{1}{2} b c \sin \alpha \end{equation*} When You Know 1 Side and 2 Angles...

Posted by
Beetle B.
on
Tue 23 November 2021

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Trigonometry and Triangles

Law of Sines Consider this triangle. Draw a line vertically down from \(B\). Label the length of this line as \(h\) (for height). Now...

Posted by
Beetle B.
on
Tue 09 November 2021

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Trigonometric Identities

Formula for \(\cos(\alpha-\beta)\) Consider the unit circle below: Here, we make \(\overline{AB}=\overline{CD}\), so they have the...

Posted by
Beetle B.
on
Mon 08 November 2021

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Binomial Expansion

Some things to note: The sum of all the coefficients is \(2^{n}\). To prove this, evaluate \(\left(a+b\right)^{n}\) with \(a=1,b=1\) The...

Posted by
Beetle B.
on
Sat 06 November 2021

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Summing the first \(n\) terms (arbitrary power)

We know that: \begin{equation*} \sum_{k=1}^{n} k=\frac{n(n+1)}{2} \end{equation*} We want to find the expression for \begin{equation*}...

Posted by
Beetle B.
on
Sat 06 November 2021

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Arithmetic, Harmonic and Geometric Means

\(A\ge G\ge H\) if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) { var align = "center", indent = "0em", linebreak =...

Posted by
Beetle B.
on
Thu 04 November 2021

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Geometric Series

The sum of a geometric sequence is: \begin{equation*} S_{n}=\frac{a\left(r^{n}-1)}{r-1} \end{equation*} Note, this is assuming \(n\)...

Posted by
Beetle B.
on
Thu 04 November 2021

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Arithmetic Series

The sum of an arithmetic series is: \begin{equation*} S=\frac{1}{2}\left(a+l\right) \end{equation*} where \(l\) is the last term. if...

Posted by
Beetle B.
on
Thu 04 November 2021

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\(z^{n}=1\)

Say we want the roots of \(z^{n}=1\). Let \(z=re^{i\theta}\). Then \(z^{n}=r^{n}e^{in\theta}=1\). We immediately see that \(r=1\). Thus...

Posted by
Beetle B.
on
Wed 03 November 2021

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Roots of Quadratic Equations

Given \(x^{2}+bx+c=0\), let the roots be \(\alpha,\beta\). Then: \begin{equation*} \alpha+\beta=-b \end{equation*} \begin{equation*}...

Posted by
Beetle B.
on
Wed 03 November 2021

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Polynomials

Remainder Theorem If a polynomial \(p(x)\) is divided by \(x-r\), then the remainder is \(p(r)\). To prove this, let \(p(x)=Q(x)(x-r) +...

Posted by
Beetle B.
on
Thu 28 October 2021

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Roots of Equations

Equations of the Form \(\sqrt{ax^{2}+bx+c}\pm\sqrt{ax^{2}+bx+d}=k\) \begin{equation*} \sqrt{ax^{2}+bx+c}+\sqrt{ax^{2}+bx+d}=k...

Posted by
Beetle B.
on
Thu 28 October 2021

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Roots of Equations In More Than One Variable

\begin{equation*} 2*x^{2}-xy+y^{2}=4 \end{equation*} \begin{equation*} 4*x^{2}-5xy+3y^{2}=6 \end{equation*} Manipulate the two equations...

Posted by
Beetle B.
on
Thu 28 October 2021

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Matrices

Determinant of a 2x2 matrix: \begin{equation*} \left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=ad-bc \end{equation*} The minor...

Posted by
Beetle B.
on
Thu 28 October 2021

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Triangle Inequality for Complex Numbers

\begin{equation*} \ ||z_{1}|-|z_{2}||\le|z_{1}+z_{2}|\le|z_{1}|+|z_{2}| \end{equation*} The second inequality follows from the triangle...

Posted by
Beetle B.
on
Mon 25 October 2021

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\(z^{n}+\overline{z}^{n}\)

\(z^{n}+\overline{z}^{n}\) is a real number for all positive \(n\). Proof 1 Treat \(z=re^{i\theta}\) and you can show that the sum...

Posted by
Beetle B.
on
Sun 24 October 2021

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Componendo/Dividendo

Given: \begin{equation*} \frac{a}{b}=\frac{c}{d} \end{equation*} Componendo: \begin{equation*} \frac{a+b}{b}=\frac{c+d}{d}...

Posted by
Beetle B.
on
Sat 23 October 2021