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

The value of sin(pi/4)sin(pi/(12)) is (sqrt(3)-1)/4

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sin(pi/4)sin(pi/(12))

Solution

sin (\frac{π}{4}) sin (\frac{π}{12})

Solution

\frac{3 - 1}{4}

Decimal Notation

0.18301 \dots

Solution steps

sin (\frac{π}{4}) sin (\frac{π}{12})

Rewrite using trig identities:

\frac{cos ( \frac{π}{6} ) - cos ( \frac{π}{3} )}{2}

sin (\frac{π}{4}) 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{π}{4} - \frac{π}{12}) - cos (\frac{π}{4} + \frac{π}{12}))

Simplify:

\frac{π}{4} - \frac{π}{12} = \frac{π}{6}

\frac{π}{4} - \frac{π}{12}

Least Common Multiplier of

4, 12 : 12

4, 12

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

4

12

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

= \frac{π 3}{12} - \frac{π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π}{12}

Add similar elements:

3 π - π = 2 π

= \frac{2 π}{12}

Cancel the common factor:

2

= \frac{π}{6}

Simplify:

\frac{π}{4} + \frac{π}{12} = \frac{π}{3}

\frac{π}{4} + \frac{π}{12}

Least Common Multiplier of

4, 12 : 12

4, 12

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

4

12

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

= \frac{π 3}{12} + \frac{π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π}{12}

Add similar elements:

3 π + π = 4 π

= \frac{4 π}{12}

Cancel the common factor:

4

= \frac{π}{3}

\frac{1}{2} (cos (\frac{π}{6}) - cos (\frac{π}{3})) = \frac{cos ( \frac{π}{6} ) - cos ( \frac{π}{3} )}{2}

\frac{1}{2} (cos (\frac{π}{6}) - cos (\frac{π}{3}))

Multiply fractions:

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

= \frac{1 \cdot ( cos ( \frac{π}{6} ) - cos ( \frac{π}{3} ) )}{2}

1 \cdot (cos (\frac{π}{6}) - cos (\frac{π}{3})) = cos (\frac{π}{6}) - cos (\frac{π}{3})

1 \cdot (cos (\frac{π}{6}) - cos (\frac{π}{3}))

Multiply:

1 \cdot (cos (\frac{π}{6}) - cos (\frac{π}{3})) = (cos (\frac{π}{6}) - cos (\frac{π}{3}))

= (cos (\frac{π}{6}) - cos (\frac{π}{3}))

Remove parentheses:

(a) = a

= cos (\frac{π}{6}) - cos (\frac{π}{3})

= \frac{cos ( \frac{π}{6} ) - cos ( \frac{π}{3} )}{2}

= \frac{cos ( \frac{π}{6} ) - cos ( \frac{π}{3} )}{2}

= \frac{cos ( \frac{π}{6} ) - cos ( \frac{π}{3} )}{2}

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

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{3}{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}

= \frac{\frac{3}{2} - \frac{1}{2}}{2}

Simplify

\frac{\frac{3}{2} - \frac{1}{2}}{2} : \frac{3 - 1}{4}

\frac{\frac{3}{2} - \frac{1}{2}}{2}

Combine the fractions

\frac{3}{2} - \frac{1}{2} : \frac{3 - 1}{2}

Apply rule

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

= \frac{3 - 1}{2}

= \frac{\frac{3 - 1}{2}}{2}

Apply the fraction rule:

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

= \frac{3 - 1}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 - 1}{4}

= \frac{3 - 1}{4}

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Frequently Asked Questions (FAQ)

What is the value of sin(pi/4)sin(pi/(12)) ?
The value of sin(pi/4)sin(pi/(12)) is (sqrt(3)-1)/4