Sign of Sine

Theorem
Let $x$ be a real number.

Then:

where $\sin$ is the real sine function and $n$ is an integer.

Proof
Proof by induction:

Base case
For $n = 0$, it follows from Sine and Cosine are Periodic on Reals/Corollary.

Induction Hypothesis
This is our induction hypothesis:

Now we need to show true for $n=k+1$:

Induction Step
This is our induction step:

Also:

The result follows by induction.

For negative $n$:

Induction Hypothesis
This is our induction hypothesis:

Now we need to show true for $n=k-1$:

Induction Step
This is our induction step:

Also:

The result follows by induction.

Also see

 * Sign of Cosine
 * Sign of Tangent
 * Sign of Cosecant
 * Sign of Secant
 * Sign of Tangent