## An Easy Way To Learn The Basic Trigonometric Identities

First learn the structure. Learn the positions of the six trigonometry functions. First comes the $\sin$ function, underneath it comes the $\cos$ function.

Then in the next column comes the $\csc$ function, underneath it comes the $\sec$ function. And in the last column comes the $\cot$ function and underneath it comes the $\tan$ function. Learning this position is important and just one time. The six trig functions are specially placed in the above given places, so as to serve our need.  » Read more

## The Trigonometric Ratio Hexagon

I found a book this morning containing a hexagonal figure of trigonometric ratios.  It reminded me of the mnemonics I created when I was still in high school in order to remember the formulas during exams. Unlike in other countries, we were not allowed to have formula sheets during exams, so we have to memorize them all.

In the figure, any trigonometric ratio is a product of its immediate neighbors. Therefore,  » Read more

## The Proof of the Tangent Half-Angle Formula

In this post, we prove the following trigonometric identity:

$\displaystyle \tan \frac{\theta}{2} = \frac{\sin\theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}$.

Proof

Consider a semi-circle with “center” $O$ and diameter $AB$ and radius equal to 1 unit as shown below.  If we let $\angle BOC =\theta$, then by the Inscribed Angle Theorem, $\angle CAB = \frac{\theta}{2}$.

Draw $CD$ perpendicular to $OB$ as shown in the second figure. We can compute for the sine and cosine of $\theta$ which equal to the lengths of $CD$ and $OD$, respectively. In effect, $BD = 1 - \cos \theta$ and $AD = 1 + \cos \theta$. » Read more

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