What is the value of sin(pi/(12))-sin((17pi)/(12)) ?

The value of sin(pi/(12))-sin((17pi)/(12)) is sqrt(3/2)

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sin(pi/(12))-sin((17pi)/(12))

Solution

sin (\frac{π}{12}) - sin (\frac{17 π}{12})

Solution

\frac{3}{2}

Decimal Notation

1.22474 \dots

Solution steps

sin (\frac{π}{12}) - sin (\frac{17 π}{12})

Rewrite using trig identities:

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

sin (\frac{π}{12}) - sin (\frac{17 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{17 π}{12}}{2}) cos (\frac{\frac{π}{12} + \frac{17 π}{12}}{2})

Simplify:

\frac{\frac{π}{12} - \frac{17 π}{12}}{2} = - \frac{2 π}{3}

\frac{\frac{π}{12} - \frac{17 π}{12}}{2}

Combine the fractions

\frac{π}{12} - \frac{17 π}{12} : - \frac{4 π}{3}

Apply rule

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

= \frac{π - 17 π}{12}

Add similar elements:

π - 17 π = - 16 π

= \frac{- 16 π}{12}

Apply the fraction rule:

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

= - \frac{16 π}{12}

Cancel the common factor:

4

= - \frac{4 π}{3}

= \frac{- \frac{4 π}{3}}{2}

Apply the fraction rule:

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

= - \frac{\frac{4 π}{3}}{2}

Apply the fraction rule:

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

\frac{\frac{4 π}{3}}{2} = \frac{4 π}{3 \cdot 2}

= - \frac{4 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= - \frac{4 π}{6}

Cancel the common factor:

2

= - \frac{2 π}{3}

Simplify:

\frac{\frac{π}{12} + \frac{17 π}{12}}{2} = \frac{3 π}{4}

\frac{\frac{π}{12} + \frac{17 π}{12}}{2}

Combine the fractions

\frac{π}{12} + \frac{17 π}{12} : \frac{3 π}{2}

Apply rule

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

= \frac{π + 17 π}{12}

Add similar elements:

π + 17 π = 18 π

= \frac{18 π}{12}

Cancel the common factor:

6

= \frac{3 π}{2}

= \frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

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

Use the following property:

sin (- x) = - sin (x)

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

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

Simplify

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

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

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 π}{3})

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

Simplify

- 2 \cdot \frac{3}{2} (- \frac{2}{2}) : \frac{3}{2}

- 2 \cdot \frac{3}{2} (- \frac{2}{2})

Apply rule

- (- a) = a

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

Cancel the common factor:

2

= \frac{3 2}{2}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{2}

= \frac{3}{2}

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Frequently Asked Questions (FAQ)

What is the value of sin(pi/(12))-sin((17pi)/(12)) ?
The value of sin(pi/(12))-sin((17pi)/(12)) is sqrt(3/2)