What is the value of sec((14pi)/3) ?

The value of sec((14pi)/3) is -2

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sec((14pi)/3)

sec (\frac{14 π}{3})

- 2

Solution steps

sec (\frac{14 π}{3})

sec (\frac{14 π}{3}) = sec (\frac{2 π}{3})

sec (\frac{14 π}{3})

Rewrite

\frac{14 π}{3}

2 π \cdot 2 + \frac{2 π}{3}

= sec (2 π 2 + \frac{2 π}{3})

Apply the periodicity of

sec

sec (x + 2 π \cdot k) = sec (x)

sec (2 π \cdot 2 + \frac{2 π}{3}) = sec (\frac{2 π}{3})

= sec (\frac{2 π}{3})

= sec (\frac{2 π}{3})

Use the following trivial identity:

sec (\frac{2 π}{3}) = - 2

sec (\frac{2 π}{3})

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= - 2

= - 2

csc (arcsec (- 2))

\frac{cos ( 6 0 ^{\circ} )}{cos ( 6 0 ^{\circ} )}

arccos (\frac{- 58}{129 53})

arctan (\frac{21}{28})

cos^{2} (3 3^{\circ}) - sin^{2} (3 3^{\circ})