# What is the domain of function f(x)={sinx}/{tanx}?

Jun 28, 2015

Domain of the function$f \left(x\right) = \sin \frac{x}{\tan} x$

#### Explanation:

The common period, or domain, of a trig function F(x), containing 2 trig functions f(x) and g(x), should be the least multiple of the 2 periods.
Here, the period of sin x is $2 \pi$, and the period of tan x is $\pi$, therefor, their common period is $2 \pi$.

Jun 28, 2015

The domain of $f$ is all real numbers except integer multiples of $\frac{\pi}{2}$. All real $x$ with $x \ne k \frac{\pi}{2}$ with $k$ an integer.

#### Explanation:

Tangent in not defined for odd multiples of $\frac{\pi}{2}$ (where the cosine is $0$.)

Tangent is $0$, at integer multiples of $\pi$, so this $f$ is not defined for integer multiples of $\pi$.

Since the even multiples of $\frac{\pi}{2}$ are the integer multiples of $\pi$
and, since we have also excluded odd multiples of $\frac{\pi}{2}$,

the domain excludes all integer multiples of $\frac{\pi}{2}$.(both even and odd)

Note
For all $x$ in the domain, this $f \left(x\right)$ simplifies to $f \left(x\right) = \cos x$.