What is the difference between "$=$" and "$\equiv$"?

Question

I was recently thinking about some of my past math classes, and depending on the context I recall my professors would sometimes use the "$\equiv$" symbol in places where I'd feel "$=$" to be more appropriate. For example, since this would often be the case in my classes on differential equations and Fourier series, we would have (for $n \in \Bbb N, k \in \Bbb Z$)

$$(-1)^{2n+1} \equiv -1$$ $$\sin(k\pi) \equiv 0$$

Is there a particular reason in this context why we would say "$\equiv$" instead of "$=$"? The latter feels more natural in this context, which makes me think that there's some reason my professors would use the former.

I'm familiar with the notion of the "$\equiv$" symbol in the context of, say, elementary number theory (specifically modular arithmetic) where we might say

$$10 \equiv 1 \pmod 3$$

which isn't saying "$10$ equals $1$", just that "$10$ is like $1$ in this context." But that doesn't seem to fit the case as with the first two statements - because I don't believe it is that $(-1)^{2n+1}$ is like $-1$, or that $\sin(k \pi)$ is like $0$, they are $-1$ and $0$ respectively.

Am I just mistaken on this latter fact? Is there something I'm missing? What, precisely, is the difference between the two notations?

Answer 1

I'll give an example of each.

$$2x=x+1$$

This holds when $x=1$ only, and so the equality symbol is appropriate. In short, we use an $=$ when specific values solve the expression.

On the contrary, we have:

$$2x\equiv x +x$$ Whatever the value of $x$, this holds. This is an algebraically obvious one, but another might be $$\sin^2 x + \cos^2 x \equiv 1$$

The identity symbol $\equiv$ is used when an equality holds for all values in the domain specified (e.g. $\Bbb R$).

Answer 2

The equal sign "$=$" is used for equalities ,equations, identities and definitions of functions.

Examples of equalities are $3=2+1$ or $12^2 = 144$ where two numbers are the same.

Examples of equations are $ 3x+1=10$ or $x^2-4=0$ where for some values of the variable both sides result in the same value.

Examples of identities are $\sin ^2 x + \cos ^2 x =1$ or $(x+y)^2 = x^2+y^2+2xy$ where both sides are identical for every value of variables.

For identities sometimes $\equiv $ is used instead of $=$ for example we may use $ e^{i\theta} \equiv \cos \theta + i \sin \theta$ or $\sin ^2 x + \cos ^2 x \equiv 1$ or $(x+y)^2 \equiv x^2+y^2+2xy$ to emphasize that this is an identity not an equation.

For functions we sometimes use $ f(x) \equiv c$ to emphasize that the given value $c$ is for all values of $x$