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

The value of sin((5pi)/(12))sin(pi/(12)) is 1/4

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

Solution

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

Solution

\frac{1}{4}

Decimal Notation

0.25

Solution steps

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

Rewrite using trig identities:

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

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

Use the Product to Sum identity:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

= \frac{1}{2} (cos (\frac{5 π}{12} - \frac{π}{12}) - cos (\frac{5 π}{12} + \frac{π}{12}))

Simplify:

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

\frac{5 π}{12} - \frac{π}{12}

Apply rule

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

= \frac{5 π - π}{12}

Add similar elements:

5 π - π = 4 π

= \frac{4 π}{12}

Cancel the common factor:

4

= \frac{π}{3}

Simplify:

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

\frac{5 π}{12} + \frac{π}{12}

Apply rule

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

= \frac{5 π + π}{12}

Add similar elements:

5 π + π = 6 π

= \frac{6 π}{12}

Cancel the common factor:

6

= \frac{π}{2}

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

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

Multiply fractions:

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

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

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

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

Multiply:

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

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

Remove parentheses:

(a) = a

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

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

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

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

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{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:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

= \frac{\frac{1}{2} - 0}{2}

Simplify

\frac{\frac{1}{2} - 0}{2} : \frac{1}{4}

\frac{\frac{1}{2} - 0}{2}

\frac{1}{2} - 0 = \frac{1}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{1}{4}

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Frequently Asked Questions (FAQ)

What is the value of sin((5pi)/(12))sin(pi/(12)) ?
The value of sin((5pi)/(12))sin(pi/(12)) is 1/4