trigonometric identities Archives

An Easy Way To Learn The Basic Trigonometric Identities

January 12, 2013 GB High School Mathematics, High School Trigonometry

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

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The Proof of the Tangent Half-Angle Formula

May 3, 2012 GB High School Mathematics, High School Trigonometry, Problem Solving and Proofs

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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