What is the value of sin(1/(\frac{2){11pi}}) ?

The value of sin(1/(\frac{2){11pi}}) is -1

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sin(1/(\frac{2){11pi}})

sin (\frac{1}{\frac{2}{11 π}})

- 1

Solution steps

sin (\frac{1}{\frac{2}{11 π}})

Rewrite using trig identities:

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

sin (\frac{1}{\frac{2}{11 π}})

Write

sin (\frac{1}{\frac{2}{11 π}})

sin (2 \cdot \frac{11 π}{4})

= sin (2 \cdot \frac{11 π}{4})

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

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

sin (\frac{11 π}{4})

Rewrite

\frac{11 π}{4}

2 π + \frac{3 π}{4}

= sin (2 π + \frac{3 π}{4})

Apply the periodicity of

sin

sin (x + 2 π) = sin (x)

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

= sin (\frac{3 π}{4})

= 2 sin (\frac{3 π}{4}) cos (\frac{11 π}{4})

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})

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

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

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:

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}

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - \frac{2 2}{2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

= - 1

sec (- 1 4^{\circ})

60 \cdot sin (4 5^{\circ})

2 cos^{2} (\frac{3 π}{4}) - 1

sec (0.64)

2 - 2 sin (\frac{π}{4})

What is the value of sin(1/(\frac{2){11pi}}) ?
The value of sin(1/(\frac{2){11pi}}) is -1