What is the value of sec((17pi)/(12)) ?

The value of sec((17pi)/(12)) is -sqrt(2)-sqrt(6)

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sec((17pi)/(12))

Solution

sec (\frac{17 π}{12})

Solution

- 2 - 6

Decimal Notation

- 3.86370 \dots

Solution steps

sec (\frac{17 π}{12})

Rewrite using trig identities:

\frac{1}{cos ( \frac{17 π}{12} )}

sec (\frac{17 π}{12})

Use the basic trigonometric identity:

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

= \frac{1}{cos ( \frac{17 π}{12} )}

= \frac{1}{cos ( \frac{17 π}{12} )}

Rewrite using trig identities:

cos (\frac{17 π}{12}) = \frac{2 - 6}{4}

cos (\frac{17 π}{12})

Rewrite using trig identities:

cos (\frac{3 π}{4}) cos (\frac{2 π}{3}) - sin (\frac{3 π}{4}) sin (\frac{2 π}{3})

cos (\frac{17 π}{12})

Write

cos (\frac{17 π}{12})

cos (\frac{3 π}{4} + \frac{2 π}{3})

= cos (\frac{3 π}{4} + \frac{2 π}{3})

Use the Angle Sum identity:

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

= cos (\frac{3 π}{4}) cos (\frac{2 π}{3}) - sin (\frac{3 π}{4}) sin (\frac{2 π}{3})

= cos (\frac{3 π}{4}) cos (\frac{2 π}{3}) - sin (\frac{3 π}{4}) sin (\frac{2 π}{3})

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{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

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{1}{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{2 π}{3}) = \frac{3}{2}

sin (\frac{2 π}{3})

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{3}{2}

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

Simplify

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

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

Remove parentheses:

(- a) = - a, - (- a) = a

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

\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{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}{4} - \frac{6}{4}

Apply rule

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

= \frac{2 - 6}{4}

= \frac{2 - 6}{4}

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

Simplify

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

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

Apply the fraction rule:

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

= \frac{4}{2 - 6}

Rationalize

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

\frac{4}{2 - 6}

Multiply by the conjugate

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

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

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

(2 - 6) (2 + 6)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 2, b = 6

= (2)^{2} - (6)^{2}

Simplify

(2)^{2} - (6)^{2} : - 4

(2)^{2} - (6)^{2}

(2)^{2} = 2

(2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

(6)^{2} = 6

(6)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 6^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 6

= 2 - 6

Subtract the numbers:

2 - 6 = - 4

= - 4

= - 4

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

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

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

Divide the numbers:

\frac{4}{4} = 1

= - (2 + 6)

Distribute parentheses

= - (2) - (6)

Apply minus-plus rules

+ (- a) = - a

= - 2 - 6

= - 2 - 6

= - 2 - 6

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Frequently Asked Questions (FAQ)

What is the value of sec((17pi)/(12)) ?
The value of sec((17pi)/(12)) is -sqrt(2)-sqrt(6)