Why is $\sin^2x + \cos^2x = 1$ important?
To start off, I understand the proof behind this identity, and I can visualize it in my head with the unit circle.
But I read this quote:
They only need to remember three facts – that $\sin 30^\circ = ½$ , that $\tan 45^\circ =1$, and that $\sin^2x + \cos^2x =1$ . Just about everything else they need to know about trigonometry can be derived from these.
and I realized I don't have a complete understanding on practical use of the identity. Therefore I am looking for an explanation and some practical examples on why it is so important.
Thanks for your help!
$\endgroup$ 22 Answers
$\begingroup$The "special angles" in trig are $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$ (and their counterparts in quadrants II, III, and IV).
Presumably, the sines and cosines at $0^\circ$ and $90^\circ$ are obvious to the quoted author, so the quotation boils-down what's needed to remember values at the remaining angles to three facts.
"$\sin^2\theta+\cos^2\theta=1$" allows for conversion between sines and cosines. Always handy.
"$\sin 30^\circ = 1/2$" immediately gives $\cos 60^\circ =1/2$. (Complementary angles are cool like that, since "cosine" means "complementary sine".) With the help of the Pythagorean identity above, we also get $\cos 30^\circ = \sin 60^\circ = \sqrt{3}/2$. (If you're going to commit any of these to memory, it makes sense to choose the simple fraction, $1/2$. Personally, I think remembering "$\cos 60^\circ = 1/2$" is easier, because I can "see" it more easily in dropping a perpendicular from the top vertex of an equilateral triangle.)
"$\tan 45^\circ=1$" encodes information about the remaining special angle, saying that (in Quadrant I) $\sin 45^\circ = \cos 45^\circ$. Again with the help of the Pythagorean identity, we have $1 = \sin^2 45^\circ + \cos^2 45^\circ = 2 \sin^2 45^\circ$, so that $\sin 45^\circ = \cos 45^\circ = 1/\sqrt{2}$.
So, the three facts help to recover a total of seven facts. And then getting the remaining secants, tangents, cosecants, and cotangents is a simple matter of using various ratios. The sine-cosine Pythagorean identity also gives rise to the others, by dividing-through by $\cos^2\theta$ or $\sin^2\theta$ ...
$$\begin{align} \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta} \quad &\to \quad \tan^2\theta + 1 = \sec^2\theta \\[6pt] \frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta} \quad &\to \quad 1 + \cot^2\theta = \csc^2\theta \end{align}$$
... reducing pressure to memorize all three relations.
There are other (and better?) ways to remember this stuff, but the basic point is that there's a good deal of informational redundancy in trig. While it may seem like there are a zillion different things to memorize, it doesn't take long to realize that one can focus on a (very) few key facts and re-derive the rest on demand.
$\endgroup$ 1 $\begingroup$It's importance is in its equivalence to Pythagoras' Theorem. It shows that the curve $(\cos t,\sin t)$ traces the unit circle. It relates values of one function in terms of the other.
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