# What is the period of f(theta) = tan ( (15 theta)/7) - cos ( (2 theta)/ 5 ) ?

Apr 23, 2016

$35 \pi$

#### Explanation:

The period of both $\sin k \theta \mathmr{and} \tan k \theta$ is $\frac{2 \pi}{k}$

Here; the periods of the separate terms are $\frac{14 \pi}{15} \mathmr{and} 5 \pi$..

The compounded period for the sum $f \left(\theta\right)$ is given by

$\left(\frac{14}{15}\right) \pi L = 5 \pi M$, for the least multiples L and Ml that get common value as an integer multiple of $\pi$..

L = 75/2 and M = 7, and the common integer value is $35 \pi$.

So, the period of $f \left(\theta\right) = 35 \pi$.

Now, see the effect of the period.

$f \left(\theta + 35 \pi\right)$

=tan((15/7)(theta+35pi))-cos((2/5)(theta+35pi))

=tan(75pi+(15/7)theta)-cos(14pi+(2/5)theta))=tan ((15/7)theta)

-cos((2/5)theta))

$= f \left(\theta\right)$

Note that 75pi+_ is in the 3rd quadrant and tangent is positive. Similarly, for the cosine, $14 \pi +$ is in the 1st quadrant and cosine is positive.

The value repeats when $\theta$ is increased by any integer multiple of $35 \pi$.