What is the value of csc((11pi)/8) ?

The value of csc((11pi)/8) is -2sqrt(2+\sqrt{2)}+sqrt(2)sqrt(2+\sqrt{2)}

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Popular Trigonometry >

csc((11pi)/8)

Solution

csc (\frac{11 π}{8})

Solution

- 2 2 + 2 + 2 2 + 2

Decimal

- 1.08239 \dots

Solution steps

csc (\frac{11 π}{8})

Rewrite using trig identities:

\frac{1}{sin ( \frac{11 π}{8} )}

csc (\frac{11 π}{8})

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( \frac{11 π}{8} )}

= \frac{1}{sin ( \frac{11 π}{8} )}

Rewrite using trig identities:

sin (\frac{11 π}{8}) = - \frac{2 + 2}{2}

sin (\frac{11 π}{8})

Rewrite using trig identities:

- \frac{1 - cos ( \frac{3 π}{4} )}{2}

sin (\frac{11 π}{8})

Write

sin (\frac{11 π}{8})

sin (\frac{\frac{11 π}{4}}{2})

= sin (\frac{\frac{11 π}{4}}{2})

Use the Half Angle identity:

sin (\frac{θ}{2}) = - \frac{1 - cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 1 - 2 sin^{2} (\frac{θ}{2})

Switch sides

2 sin^{2} (\frac{θ}{2}) = 1 - cos (θ)

Divide both sides by

2

sin^{2} (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, \frac{π}{2}] [\frac{π}{2}, π] [π, \frac{3 π}{2}] [\frac{3 π}{2}, 2 π] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

sin (\frac{θ}{2}) = - \frac{( 1 - cos ( θ ) )}{2}

= - \frac{1 - cos ( \frac{11 π}{4} )}{2}

cos (\frac{11 π}{4}) = cos (\frac{3 π}{4})

cos (\frac{11 π}{4})

Rewrite

\frac{11 π}{4}

2 π + \frac{3 π}{4}

= cos (2 π + \frac{3 π}{4})

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

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

= cos (\frac{3 π}{4})

= - \frac{1 - cos ( \frac{3 π}{4} )}{2}

= - \frac{1 - cos ( \frac{3 π}{4} )}{2}

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}

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

Simplify

- \frac{1 - ( - \frac{2}{2} )}{2} : - \frac{2 + 2}{2}

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

Apply rule

- (- a) = a

= - \frac{1 + \frac{2}{2}}{2}

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

\frac{1 + \frac{2}{2}}{2}

Join

1 + \frac{2}{2} : \frac{2 + 2}{2}

1 + \frac{2}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 2}{2}

= \frac{\frac{2 + 2}{2}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 + 2}{4}

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

Simplify

\frac{2 + 2}{4} : \frac{2 + 2}{2}

\frac{2 + 2}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 + 2}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 + 2}{2}

= - \frac{2 + 2}{2}

= - \frac{2 + 2}{2}

= \frac{1}{- \frac{2 + 2}{2}}

Simplify

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

\frac{1}{- \frac{2 + 2}{2}}

Apply the fraction rule:

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

= - \frac{1}{\frac{2 + 2}{2}}

Apply the fraction rule:

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

\frac{1}{\frac{2 + 2}{2}} = \frac{2}{2 + 2}

= - \frac{2}{2 + 2}

Rationalize

- \frac{2}{2 + 2} : 2 2 + 2 - 2 2 + 2

- \frac{2}{2 + 2}

Multiply by the conjugate

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

= - \frac{2 2 + 2}{2 + 2 2 + 2}

2 + 2 2 + 2 = 2 + 2

2 + 2 2 + 2

Apply radical rule:

a a = a

2 + 2 2 + 2 = 2 + 2

= 2 + 2

= - \frac{2 2 + 2}{2 + 2}

Cancel

\frac{2 2 + 2}{2 + 2} : \frac{2 2 + 2}{1 + 2}

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

Factor

2 + 2 : 2 (2 + 1)

2 + 2

2 = 22

= 22 + 2

Factor out common term

2

= 2 (2 + 1)

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

Cancel

\frac{2 2 + 2}{2 ( 2 + 1 )} : \frac{2 2 + 2}{2 + 1}

\frac{2 2 + 2}{2 ( 2 + 1 )}

Apply radical rule:

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

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

Apply exponent rule:

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

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

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 2}{2 + 1}

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{2 2 + 2}{2 + 1}

= \frac{2 2 + 2}{2 + 1}

= - \frac{2 2 + 2}{1 + 2}

Multiply by the conjugate

\frac{2 - 1}{2 - 1}

= - \frac{2 2 + 2 ( 2 - 1 )}{( 2 + 1 ) ( 2 - 1 )}

2 2 + 2 (2 - 1) = 2 2 + 2 - 2 2 + 2

2 2 + 2 (2 - 1)

= 2 (2 - 1) 2 + 2

Apply the distributive law:

a (b - c) = ab - a c

a = 2 2 + 2, b = 2, c = 1

= 2 2 + 2 2 - 2 2 + 2 \cdot 1

= 22 2 + 2 - 1 \cdot 2 2 + 2

Simplify

22 2 + 2 - 1 \cdot 2 2 + 2 : 2 2 + 2 - 2 2 + 2

22 2 + 2 - 1 \cdot 2 2 + 2

Apply radical rule:

a a = a

22 = 2

= 2 2 + 2 - 1 \cdot 2 2 + 2

Multiply:

1 \cdot 2 = 2

= 2 2 + 2 - 2 2 + 2

= 2 2 + 2 - 2 2 + 2

(2 + 1) (2 - 1) = 1

(2 + 1) (2 - 1)

Apply Difference of Two Squares Formula:

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

a = 2, b = 1

= (2)^{2} - 1^{2}

Simplify

(2)^{2} - 1^{2} : 1

(2)^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (2)^{2} - 1

(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

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= 1

= - \frac{2 2 + 2 - 2 2 + 2}{1}

Apply rule

\frac{a}{1} = a

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

Distribute parentheses

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

Apply minus-plus rules

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

= - 2 2 + 2 + 2 2 + 2

= - 2 2 + 2 + 2 2 + 2

= - 2 2 + 2 + 2 2 + 2

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Frequently Asked Questions (FAQ)

What is the value of csc((11pi)/8) ?
The value of csc((11pi)/8) is -2sqrt(2+\sqrt{2)}+sqrt(2)sqrt(2+\sqrt{2)}