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

The value of sin(pi/9)cos((2pi)/9)+cos(pi/9)sin((2pi)/9) is (sqrt(3))/2

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

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

\frac{3}{2}

Decimal Notation

0.86602 \dots

Solution steps

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

Rewrite using trig identities:

sin (\frac{π}{3})

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

Use the Angle Sum identity:

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

= sin (\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}

= sin (\frac{π}{3})

= sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{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}

= \frac{3}{2}

cot (6 0^{\circ}) tan (3 0^{\circ}) + sec^{2} (4 5^{\circ})

arctan (\frac{20}{40})

110 cos (15 0^{\circ})

cos^{2} (27 0^{\circ})

2 cos^{2} (\frac{π}{5}) - 1

What is the value of sin(pi/9)cos((2pi)/9)+cos(pi/9)sin((2pi)/9) ?
The value of sin(pi/9)cos((2pi)/9)+cos(pi/9)sin((2pi)/9) is (sqrt(3))/2