What is the value of sec((-7pi)/3) ?

The value of sec((-7pi)/3) is 2

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sec((-7pi)/3)

sec (\frac{- 7 π}{3})

2

Solution steps

sec (\frac{- 7 π}{3})

Simplify

= sec (- \frac{7 π}{3})

Use the following property:

sec (- x) = sec (x)

sec (- \frac{7 π}{3}) = sec (\frac{7 π}{3})

= sec (\frac{7 π}{3})

sec (\frac{7 π}{3}) = sec (\frac{π}{3})

sec (\frac{7 π}{3})

Rewrite

\frac{7 π}{3}

2 π + \frac{π}{3}

= sec (2 π + \frac{π}{3})

Apply the periodicity of

sec

sec (x + 2 π) = sec (x)

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

= sec (\frac{π}{3})

= sec (\frac{π}{3})

Use the following trivial identity:

sec (\frac{π}{3}) = 2

sec (\frac{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

128 + 127.5 sin (\frac{2 π}{16} \cdot 9)

128 + 127.5 sin (\frac{2 π}{16} \cdot 8)

15 sin (1 0^{\circ})

128 + 127.5 sin (\frac{2 π}{16} \cdot 2)

128 + 127.5 sin (\frac{2 π}{16} \cdot 5)