What is the value of sec(-(11pi)/(12)) ?

The value of sec(-(11pi)/(12)) is -sqrt(6)+sqrt(2)

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sec(-(11pi)/(12))

Solution

sec (- \frac{11 π}{12})

Solution

- 6 + 2

Decimal Notation

- 1.03527 \dots

Solution steps

sec (- \frac{11 π}{12})

Use the following property:

sec (- x) = sec (x)

sec (- \frac{11 π}{12}) = sec (\frac{11 π}{12})

= sec (\frac{11 π}{12})

Rewrite using trig identities:

\frac{1}{cos ( \frac{11 π}{12} )}

sec (\frac{11 π}{12})

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( \frac{11 π}{12} )}

= \frac{1}{cos ( \frac{11 π}{12} )}

Rewrite using trig identities:

cos (\frac{11 π}{12}) = \frac{- 6 - 2}{4}

cos (\frac{11 π}{12})

Rewrite using trig identities:

cos (\frac{3 π}{4}) cos (\frac{π}{6}) - sin (\frac{3 π}{4}) sin (\frac{π}{6})

cos (\frac{11 π}{12})

Write

cos (\frac{11 π}{12})

cos (\frac{3 π}{4} + \frac{π}{6})

= cos (\frac{3 π}{4} + \frac{π}{6})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (\frac{3 π}{4}) cos (\frac{π}{6}) - sin (\frac{3 π}{4}) sin (\frac{π}{6})

= cos (\frac{3 π}{4}) cos (\frac{π}{6}) - sin (\frac{3 π}{4}) sin (\frac{π}{6})

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

sin (\frac{3 π}{4})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= (- \frac{2}{2}) \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

Simplify

(- \frac{2}{2}) \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2} : \frac{- 6 - 2}{4}

(- \frac{2}{2}) \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

Remove parentheses:

(- a) = - a

= - \frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{2 3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 3}{4}

Simplify

23 : 6

23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

\frac{2}{2} \cdot \frac{1}{2} = \frac{2}{4}

\frac{2}{2} \cdot \frac{1}{2}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{2 \cdot 1}{2 \cdot 2}

Multiply:

2 \cdot 1 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

= - \frac{6}{4} - \frac{2}{4}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 6 - 2}{4}

= \frac{- 6 - 2}{4}

= \frac{1}{\frac{- 6 - 2}{4}}

Simplify

\frac{1}{\frac{- 6 - 2}{4}} : - 6 + 2

\frac{1}{\frac{- 6 - 2}{4}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{4}{- 6 - 2}

Rationalize

\frac{4}{- 6 - 2} : 2 - 6

\frac{4}{- 6 - 2}

Multiply by the conjugate

\frac{- 6 + 2}{- 6 + 2}

= \frac{4 ( - 6 + 2 )}{( - 6 - 2 ) ( - 6 + 2 )}

(- 6 - 2) (- 6 + 2) = 4

(- 6 - 2) (- 6 + 2)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = - 6, b = - 2, c = - 6, d = 2

= (- 6) (- 6) + (- 6) 2 + (- 2) (- 6) + (- 2) 2

Apply minus-plus rules

(- a) (- b) = ab, + (- a) = - a

= 66 - 62 + 26 - 22

Simplify

66 - 62 + 26 - 22 : 4

66 - 62 + 26 - 22

Add similar elements:

- 62 + 26 = 0

= 66 - 22

Apply radical rule:

a a = a

66 = 6

= 6 - 22

Apply radical rule:

a a = a

22 = 2

= 6 - 2

Subtract the numbers:

6 - 2 = 4

= 4

= 4

= \frac{4 ( - 6 + 2 )}{4}

Divide the numbers:

\frac{4}{4} = 1

= - 6 + 2

= - 6 + 2

= - 6 + 2

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Frequently Asked Questions (FAQ)

What is the value of sec(-(11pi)/(12)) ?
The value of sec(-(11pi)/(12)) is -sqrt(6)+sqrt(2)