Category Archives: Problem Solving and Proofs

The Proof of the Tangent Half-Angle Formula

Posted on May 3, 2012 by Guillermo Bautista | 1 Comment

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 $AD = 1 + \cos \theta$ . Continue reading →

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Posted in High School Mathematics, High School Trigonometry, Problem Solving and Proofs

Tagged half angle formula, half angle formula identity, half angle formula proof, inscribed angle theorem, proof without words, trigonometric identities, trigonometric identity

Limit by epsilon-delta proof: Example 2

Posted on November 1, 2011 by Guillermo Bautista | Leave a comment

This is the overdelayed continuation of the discussion on the $\epsilon-\delta$ definition of limits. In this post, we discuss another example.

Prove that the $\lim_{x \to 2} x^2 = 4$ .

Recall that the definition states that the limit of $f(x) = L$ as $x$ approaches $a$ if for all $\epsilon > 0$ , however small, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$ , then $|f(x) - L| < \epsilon$ .

From the example 1, we have learned that we should manipulate $|f(x)-L=|x^2 - 4|$ , to make one of the expressions look like $|x-a|=|x-2|$ . Solving, we have

$|f(x) - L| = |x^2 - 4| = |(x+2)(x-2)| = |x+2||x-2|$ .

Note that we have accomplished our goal, going back to the definition, this means that if $0 < x - 2 < \delta$ , then $|x+2||x-2| < \epsilon$ .

Now, it is not possible to divide both sides by $x + 2$ (making it $|x-2| < \frac{\epsilon}{|x+2|})$ because $x$ varies. This means that we have to find a constant $k$ such that $|x + 2| < k$ . Continue reading →

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Posted in Calculus and Analysis, Problem Solving and Proofs

Tagged calculus limits, epsilon-delta definition, limits by definition, proof of limits by definition

More examples of proof by contradiction

Posted on July 1, 2011 by Guillermo Bautista | 3 Comments

We have had good discussions on mathematical proofs, so I am planning to create a mathematical proof series that will discuss the basics such as direct proof, indirect proof, and proof by mathematical induction. But before I do that, let me continue with more examples of proof by contradiction.

Proof by contradiction, as we have discussed, is a proof strategy where you assume the opposite of a statement, and then find a contradiction somewhere in your proof. Finding a contradiction means that your assumption is false and therefore the statement is true. Below are several more examples of this proof strategy.

Example 1: $\sqrt{2}$ irrational.