What is the value of sec((19pi)/4) ?

The value of sec((19pi)/4) is -sqrt(2)

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sec((19pi)/4)

sec (\frac{19 π}{4})

- 2

Decimal Notation

- 1.41421 \dots

Solution steps

sec (\frac{19 π}{4})

sec (\frac{19 π}{4}) = sec (\frac{3 π}{4})

sec (\frac{19 π}{4})

Rewrite

\frac{19 π}{4}

2 π \cdot 2 + \frac{3 π}{4}

= sec (2 π 2 + \frac{3 π}{4})

Apply the periodicity of

sec

sec (x + 2 π \cdot k) = sec (x)

sec (2 π \cdot 2 + \frac{3 π}{4}) = sec (\frac{3 π}{4})

= sec (\frac{3 π}{4})

= sec (\frac{3 π}{4})

Use the following trivial identity:

sec (\frac{3 π}{4}) = - 2

sec (\frac{3 π}{4})

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

3 sin (18 0^{\circ})

\frac{1}{tan ( 6 0 ^{\circ} )}

(1 + cos (20 0^{\circ})) (1 + cos (34 0^{\circ}))

cos (52. 5^{\circ}) sin (7. 5^{\circ})

cot^{2} (3 0^{\circ})