What is the period of $f (θ) = tan (\frac{15 θ}{7}) - sec (\frac{5 θ}{6})$ $f(theta) = tan ( ( 15 theta)/7 )- sec ( ( 5 theta)/ 6 )$ ?

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What is the period of $f (θ) = tan (\frac{15 θ}{7}) - sec (\frac{5 θ}{6})$ $f(theta) = tan ( ( 15 theta)/7 )- sec ( ( 5 theta)/ 6 )$ ?

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Answer 1 · 2016-07-10T18:03:14

Period $P = \frac{84 π}{5} = 52.77875658$ $P=(84pi)/5=52.77875658$

Explanation:

The given $f (θ) = tan (\frac{15 θ}{7}) - sec (\frac{5 θ}{6})$ $f(theta)=tan((15theta)/7)-sec ((5theta)/6)$

For $tan (\frac{15 θ}{7})$ $tan ((15theta)/7)$ , period $P_{t} = \frac{π}{\frac{15}{7}} = \frac{7 π}{15}$ $P_t=pi/(15/7)=(7pi)/15$

For $sec (\frac{5 θ}{6})$ $sec ((5theta)/6)$ , period $P_{s} = \frac{2 π}{\frac{5}{6}} = \frac{12 π}{5}$ $P_s=(2pi)/(5/6)=(12pi)/5$

To get the period of $f (θ) = tan (\frac{15 θ}{7}) - sec (\frac{5 θ}{6})$ $f(theta)=tan((15theta)/7)-sec ((5theta)/6)$ ,
We need to obtain the LCM of the $P_{t}$ $P_t$ and $P_{s}$ $P_s$

The solution

Let $P$ $P$ be the required period
Let $k$ $k$ be an integer such that $P = k \cdot P_{t}$ $P=k*P_t$
Let $m$ $m$ be an integer such that $P = m \cdot P_{s}$ $P=m*P_s$

$P = P$ $P=P$
$k \cdot P_{t} = m \cdot P_{s}$ $k*P_t=m*P_s$
$k \cdot \frac{7 π}{15} = m \cdot \frac{12 π}{5}$ $k*(7pi)/15=m*(12pi)/5$

Solving for $\frac{k}{m}$ $k/m$

$\frac{k}{m} = \frac{15 (12) π}{5 (7) π}$ $k/m=(15(12)pi)/(5(7)pi)$

$\frac{k}{m} = \frac{36}{7}$ $k/m=36/7$

We use $k = 36$ $k=36$ and $m = 7$ $m=7$
so that
$P = k \cdot P_{t} = 36 \cdot \frac{7 π}{15} = \frac{84 π}{5}$ $P=k*P_t=36*(7pi)/15=(84pi)/5$

also

$P = m \cdot P_{s} = 7 \cdot \frac{12 π}{5} = \frac{84 π}{5}$ $P=m*P_s=7*(12pi)/5=(84pi)/5$

Period $P = \frac{84 π}{5} = 52.77875658$ $P=(84pi)/5=52.77875658$

Kindly see the graph and observe two points to verify for the period

![Desmos.com]( useruploads.socratic.org )

God bless....I hope the explanation is useful

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What is the period of $f (θ) = tan (\frac{15 θ}{7}) - sec (\frac{5 θ}{6})$ $f(theta) = tan ( ( 15 theta)/7 )- sec ( ( 5 theta)/ 6 )$ ?

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Explanation:

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What is the period of $f (θ) = tan (\frac{15 θ}{7}) - sec (\frac{5 θ}{6})$ $f(theta) = tan ( ( 15 theta)/7 )- sec ( ( 5 theta)/ 6 )$ ?