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

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

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cos((5pi)/(12))-cos(pi/(12))

Solution

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

Solution

- \frac{2}{2}

Decimal Notation

- 0.70710 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Sum to Product identity:

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

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

Simplify:

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

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

Combine the fractions

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

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{\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}

Simplify:

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

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

Combine the fractions

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

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}

= \frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

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

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

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

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{2}{2}

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}

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply:

2 \cdot 1 = 2

= - \frac{2}{2}

= - \frac{2}{2}

Popular Examples

csc((5pi)/(12))

csc (\frac{5 π}{12})

tan(45+30)

tan (4 5^{\circ} + 3 0^{\circ})

tan(91)

tan (9 1^{\circ})

arctan((sqrt(2))/(-sqrt(2)))

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

ln(cos(pi/2))

ln (cos (\frac{π}{2}))

Frequently Asked Questions (FAQ)

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