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I don't understand how to do maths, mostly because I don't understand why formulae work they way they do, or the reasoning behind equations, etc.

I tried to explain the $\sin(2\theta)$ double-angle identity to myself but failed:

Hypothetically if:

$$\text{opp} = 1 \qquad \text{adj} = 2 \qquad \text{hyp} = 3$$

then

$$\begin{align*} \sin(2\theta) &= 2\sin(\theta)\cos(\theta)\\\\ \left(\frac{\text{opp}}{\text{hyp}}\right)\cdot 2 &= 2\cdot\left(\frac{\text{opp}}{\text{hyp}}\right)\left(\frac{\text{adj}}{\text{hyp}}\right)\\\\ \frac{1}{3}\cdot 2 & = 2\left(\frac{1}{3}\right)\left(\frac{2}{3}\right)\\\\ \frac{2}{3} &\neq \frac{4}{9} \end{align*}$$

Where did I go wrong? How do the double-angle identities work?

Zev Chonoles
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Vesta
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2 Answers2

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The best way to see how the identities work is to see WHY they work. I find that formulas are much more illuminating when one sees a proof. This will, in trigonometry, usually appeal to some geometric intuition while giving you a general formula. So, we can try to prove an identity such as $$\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a).$$ Then, using this general formula, what would we know about $\sin(2a)=\sin(a+a)$?

For a proof of the general angle sum formula, here is a fairly nice geometric approach which you may find illuminating.

Erik Holmes
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Also, a right triangle with those sides does not exist due to Pythagorean Theorem.

If you need to check the formula then take for instance $\theta = \pi /2$. Then $\sin(2\theta) = \sin(\pi) = 0$. On the other hand, $2\sin(\theta)\cos(\theta)= 2 \cdot 1 \cdot 0=0$.

user 1987
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  • Alright, but could you explain how the identities work, preferably using mathematical examples? Thanks! – Vesta May 28 '15 at 04:48
  • O' my brain finally caught on with what you said facepalms.. Thanks for your help! – Vesta May 28 '15 at 05:45