# Tips For Hand Coding MathML by Stephen Bucaro

$\mathrm{sin}\theta = \sum _{n\ge 0}\frac{{\left(-1\right)}^{n\theta 2n+1}}{\left(2n+1\right)!}$

You my recognize the function shown above as the trigonometric formula to calculate arc length in radians. In this article I explain my method for hand-coding complex mathematical expressions such as this.

$\left(-1\right)n\theta 2n+1$
$<mo>(</mo><mn>-1</mn><mo>)</mo> <sup><mrow><mi>n</mi><mo>&#x03B8</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></sup>$


First, write the code for the numerator of the fraction, which contains a complex exponent.

$\left(2n+1\right)!$
$<mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>!</mo>$


Next, write the code for the denominator of the fraction.

$\frac{{\left(-1\right)}^{n\theta 2n+1}}{\left(2n+1\right)!}$
$<mfrac> <mrow> <msup> <mrow><mo>(</mo><mn>-1</mn><mo>)</mo></mrow> <mrow><mi>n</mi><mo>&#x03B8</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow> </msup> </mrow> <mrow> <mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>!</mo> </mrow> </mfrac>$


Then use an <mfrac> element to create the fraction.

$\sum \frac{{\left(-1\right)}^{n\theta 2n+1}}{\left(2n+1\right)!}$
$<mo>&#x2211;</mo> <mfrac> <mrow> <msup> <mrow><mo>(</mo><mn>-1</mn><mo>)</mo></mrow> <mrow><mi>n</mi><mo>&#x03B8</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow> </msup> </mrow> <mrow> <mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>!</mo> </mrow> </mfrac>$


Next place the code for the summation symbol before the fraction.

$\sum _{n\ge 0}\frac{{\left(-1\right)}^{n\theta 2n+1}}{\left(2n+1\right)!}$
$<munder> <mrow><ms lquote=" " rquote=" ">∑</ms></mrow> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow> </munder> <mfrac> <mrow> <msup> <mrow><mo>(</mo><mn>-1</mn><mo>)</mo></mrow> <mrow><mi>n</mi><mo>θ</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow> </msup> </mrow> <mrow> <mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>!</mo> </mrow> </mfrac>$


Then use a <munder> element to put the range under the summation symbol.

$\mathrm{sin}\theta = \sum _{n\ge 0}\frac{{\left(-1\right)}^{n\theta 2n+1}}{\left(2n+1\right)!}$
$<mi>sin</mi> <mo>θ</mo> <mo>=</mo> <munder> <mrow><ms lquote=" " rquote=" ">∑</ms></mrow> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow> </munder> <mfrac> <mrow> <msup> <mrow><mo>(</mo><mn>-1</mn><mo>)</mo></mrow> <mrow><mi>n</mi><mo>θ</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow> </msup> </mrow> <mrow> <mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>!</mo> </mrow> </mfrac>$


Then code the left side and equals sign of the equation.

$\mathrm{sin}\theta = \sum _{n\ge 0}\frac{{\left(-1\right)}^{n\theta 2n+1}}{\left(2n+1\right)!}$
$<mi>sin</mi> <mo>θ</mo> <mo>=</mo> <munder> <mrow><ms lquote=" " rquote=" ">∑</ms></mrow> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow> </munder> <mfrac> <mrow> <msup> <mrow><mo>(</mo><mn>-1</mn><mo>)</mo></mrow> <mrow> <mstyle mathsize="big"> <mi>n</mi><mo>θ</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow> </mstyle> </msup> </mrow> <mrow> <mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>!</mo> </mrow> </mfrac>$


You may have noticed that the text size of the exponent in the numerator of the fraction is very tiny. You can fix this by using an <mstyle> element with a mathsize attribute set to "big".