What is the value of cos(pi/9)cos((2pi)/9)-sin(pi/9)sin((2pi)/9) ?

The value of cos(pi/9)cos((2pi)/9)-sin(pi/9)sin((2pi)/9) is 1/2

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cos(pi/9)cos((2pi)/9)-sin(pi/9)sin((2pi)/9)

cos (\frac{π}{9}) cos (\frac{2 π}{9}) - sin (\frac{π}{9}) sin (\frac{2 π}{9})

\frac{1}{2}

Decimal Notation

0.5

Solution steps

cos (\frac{π}{9}) cos (\frac{2 π}{9}) - sin (\frac{π}{9}) sin (\frac{2 π}{9})

Rewrite using trig identities:

cos (\frac{π}{3})

cos (\frac{π}{9}) cos (\frac{2 π}{9}) - sin (\frac{π}{9}) sin (\frac{2 π}{9})

Use the Angle Sum identity:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

= cos (\frac{π}{9} + \frac{2 π}{9})

Simplify:

\frac{π}{9} + \frac{2 π}{9} = \frac{π}{3}

\frac{π}{9} + \frac{2 π}{9}

Apply rule

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

= \frac{π + 2 π}{9}

Add similar elements:

π + 2 π = 3 π

= \frac{3 π}{9}

Cancel the common factor:

3

= \frac{π}{3}

= cos (\frac{π}{3})

= cos (\frac{π}{3})

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{1}{2}

csc (3 9^{\circ})

24 tan (7 0^{\circ})

arctan (\frac{60}{40})

sin (\frac{5 π}{5})

2 π - sin (2 π)

What is the value of cos(pi/9)cos((2pi)/9)-sin(pi/9)sin((2pi)/9) ?
The value of cos(pi/9)cos((2pi)/9)-sin(pi/9)sin((2pi)/9) is 1/2