How do you build Intuition for trig identities?

Question

There are so many trigonometry identities and I understand how to apply them and even derive many of them. However, I never really have a clear understanding of what's going on. I wondered if most people take these identities for granted after they have proven them. Or how do you build intuition for dealing with these?

Answer 1

My intuition comes from different places for different identities:

1. $\sin^2t+\cos^2t=1$:

For this identity: I tend to think of the definition of $\sin$ and $\cos$ which tells us that if you start at $(1,0)$ in the plane and walk along the unit circle ($x^2+y^2=1$) a distance $t$, you arrive at $(\cos t,\sin t)$. From this point of view, this identity is visceral.

Identities like $\sec^2t=1+\tan^2t$ fall into the same category.

2. Addition laws:

For these, my intuition is that we can use Euler's formula. The exponential function has a much simpler 'addition formula': $$e^{x+y}=e^xe^y$$ Those for the trig functions are simple consequences of this one, e.g. \begin{align*} \cos(x+y)&=\Re (e^{i(x+y)})\\ &=\Re (e^{ix}e^{iy})\\ &=\Re (e^{ix})\Re (e^{iy})-\Im (e^{ix})\Im (e^{iy})\\ &=\cos x \cos y - \sin x \sin y \end{align*}

Alternately, I like to think of the $2d$ rotation matrices $$R(t)= \begin{pmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{pmatrix} $$ The addition laws are equivalent to the natural formula $R(s)R(t)=R(s+t)$ in much the same way.