

Ask HN: Intuitive formulation of trigonometry? - haliax

Hi again HN,<p>In keeping with my theme of math questions, I ask you, is there an intuitive formulation/explanation of the rules of trigonometry?<p>I understand the basics (def of sin/cos, sin^2+cos^2=1, etc.) in terms of the unit circle, but as soon as you get to things like double angle, half angle, the other identities, estimating the functions, and so on, you've lost me totally, because the proofs I've seen of those are just random things based on triangles and then you just memorize the formula and are done with it. Unfortunately, I <i>hate</i> memorizing things without being able to at least grasp the derivation from first principles. Any references?
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hackerblues
My best trick for being able to derive trig identies from scratch is to use
the complex analysis identity:

exp{i \theta} = cos(\theta) + i sin{\theta}

where i^2 = -1.

As an example, lets derive the double angle formula:

cos(2 _\theta) + i sin{2_ \theta} = exp{i 2 _\theta}

= (exp{i \theta})^2

= (cos(\theta) + i sin{\theta})^2

= cos(\theta)^2 + i 2 sin(\theta) cos(\theta) + i^2 sin{\theta}^2

= cos(\theta)^2 - sin{\theta}^2 + i 2 sin(\theta) cos(\theta)

Since the real part must equal the real part and the imaginary part equal the
imaginary part we conclude that:

cos(2_\theta) = cos(\theta)^2 - sin{\theta}^2

and

sin{2*\theta} = 2 sin(\theta) cos(\theta)

~~~

The general procedure is:

cos(complicated) = ???

1) cos(complicated) + i sin(complicated) = exp{i complicated}

2) Break it up: exp{i complicated} = exp{i simple1 + ... + i simple4} = exp{i
simple1}...exp{i simple4}

3) Translate back to sins and cos: (cos(simple1) + i
sin(simple1))...(cos(simple4) + i sin(simple4))

4) Then multiply them out and choose the real part to get cos(complicated) or
the imaginary part to get sin(complicated)

where the simple1,...,simple4 was an arbitrarily chosen number of simple bits.

