What is the value of sin(11/2 pi) ?

The value of sin(11/2 pi) is sin((3pi)/2)

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sin(11/2 pi)

sin (\frac{11}{2} π)

sin (\frac{3 π}{2})

Decimal Notation

- 1

Solution steps

sin (\frac{11}{2} π)

Rewrite as

= sin (2 \cdot \frac{11 π}{4})

Rewrite using trig identities:

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

= sin (\frac{11}{2} π)

Simplify:

\frac{11}{2} π = \frac{11 π}{2}

\frac{11}{2} π

Multiply fractions:

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

= \frac{11 π}{2}

= sin (\frac{11 π}{2})

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

sin (\frac{11 π}{2})

Rewrite

\frac{11 π}{2}

2 π \cdot 2 + \frac{3 π}{2}

= sin (2 π 2 + \frac{3 π}{2})

Apply the periodicity of

sin

sin (x + 2 π \cdot k) = sin (x)

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

= sin (\frac{3 π}{2})

= sin (\frac{3 π}{2})

cot (85 5^{\circ})

csc (\frac{17 π}{2})

arctan (\frac{- 11}{2})

- arcsin (\frac{1}{4})

14 \cdot sin (3 0^{\circ})