Month: September 2011

September 2011 Week 1 Posts Summary

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Mathematics and Multimedia

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Geometer’s Sketchpad Essentials 5 – Basic Graphing

Geometer's Sketchpad, Software Tutorials

This is the fifth part of the Geometer’s Sketchpad Essentials Series.  In this post, we are going to learn how graph using Geometer’s Sketchpad. We are going to plot the function f(x) = x^2 and g(x) = x + 1, change their properties such as colors, and thickness, and plot their intersections.

The output of our tutorial is shown above. To construct the graph, follow the step by step  instructions below.  » Read more

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The Complements Theorem

High School Mathematics, High School Trigonometry

In this post, we discuss the proof behind one of the most commonly used identities in trigonometry. We examine the equations below and  show why the relationships always hold.

\sin \theta = \cos (90^{\circ} - \theta)
\cos \theta = \sin(90^{\circ} - \theta)

To students who have taken trigonometry, I’m sure that you have met these equation before.  The proof of these equations are as follows.

Consider triangle ABC right angled at C. From the definitions, we know that

\sin A = \displaystyle\frac{a}{c}
\cos A = \displaystyle\frac{b}{c}
\sin B = \displaystyle\frac{b}{c}
\cos B = \displaystyle\frac{a}{c}

 Therefore, (1) \sin A = \cos B and (2) \cos A = \sin B.

Now, If we let A = \theta, then B = 90^{\circ} - \theta, then substituting the values of A and B in (1) and (2), we have

\sin \theta = \cos (90^{\circ} - \theta)
\cos \theta = \sin (90^{\circ} - \theta)

 and these are what we want to show.

As exercises, use the strategy above, or any strategy you want to prove the following identities.

1.) \cot\theta = \tan (90^{\circ} - \theta)
2.) \sec \theta = \csc(90^{\circ} - \theta)

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