What is the value of sin(pi/(12))-sin(5 pi/(12)) ?

The value of sin(pi/(12))-sin(5 pi/(12)) is -(sqrt(2))/2

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sin(pi/(12))-sin(5 pi/(12))

Solution

sin (\frac{π}{12}) - sin (5 \frac{π}{12})

Solution

- \frac{2}{2}

Decimal Notation

- 0.70710 \dots

Solution steps

sin (\frac{π}{12}) - sin (5 \cdot \frac{π}{12})

Simplify:

5 \cdot \frac{π}{12} = \frac{5 π}{12}

5 \cdot \frac{π}{12}

Multiply fractions:

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

= \frac{π 5}{12}

= sin (\frac{π}{12}) - sin (\frac{5 π}{12})

Rewrite using trig identities:

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

sin (\frac{π}{12}) - sin (\frac{5 π}{12})

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 2 sin (\frac{\frac{π}{12} - \frac{5 π}{12}}{2}) cos (\frac{\frac{π}{12} + \frac{5 π}{12}}{2})

Simplify:

\frac{\frac{π}{12} - \frac{5 π}{12}}{2} = - \frac{π}{6}

\frac{\frac{π}{12} - \frac{5 π}{12}}{2}

Combine the fractions

\frac{π}{12} - \frac{5 π}{12} : - \frac{π}{3}

Apply rule

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

= \frac{π - 5 π}{12}

Add similar elements:

π - 5 π = - 4 π

= \frac{- 4 π}{12}

Apply the fraction rule:

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

= - \frac{4 π}{12}

Cancel the common factor:

4

= - \frac{π}{3}

= \frac{- \frac{π}{3}}{2}

Apply the fraction rule:

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

= - \frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

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

= - \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= - \frac{π}{6}

Simplify:

\frac{\frac{π}{12} + \frac{5 π}{12}}{2} = \frac{π}{4}

\frac{\frac{π}{12} + \frac{5 π}{12}}{2}

Combine the fractions

\frac{π}{12} + \frac{5 π}{12} : \frac{π}{2}

Apply rule

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

= \frac{π + 5 π}{12}

Add similar elements:

π + 5 π = 6 π

= \frac{6 π}{12}

Cancel the common factor:

6

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

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

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{6}) = - sin (\frac{π}{6})

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

Simplify

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

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

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}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{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{1}{2} \cdot \frac{2}{2}

Simplify

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

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

Cancel the common factor:

2

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

Multiply:

1 \cdot 2 = 2

= - \frac{2}{2}

= - \frac{2}{2}

Popular Examples

sin((-7pi)/(12))

sin (\frac{- 7 π}{12})

cot((15pi)/4)

cot (\frac{15 π}{4})

-2cos(2pi)

- 2 cos (2 π)

cos((6pi)/5)

cos (\frac{6 π}{5})

-2sin(2pi)

- 2 sin (2 π)

Frequently Asked Questions (FAQ)

What is the value of sin(pi/(12))-sin(5 pi/(12)) ?
The value of sin(pi/(12))-sin(5 pi/(12)) is -(sqrt(2))/2