What is the value of cos(pi/4+(2pi)/3) ?

The value of cos(pi/4+(2pi)/3) is (sqrt(2)(-1-sqrt(3)))/4

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cos(pi/4+(2pi)/3)

cos (\frac{π}{4} + \frac{2 π}{3})

\frac{2 ( - 1 - 3 )}{4}

Decimal Notation

- 0.96592 \dots

Solution steps

cos (\frac{π}{4} + \frac{2 π}{3})

Rewrite using trig identities:

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

cos (\frac{2 π}{3} + \frac{π}{4})

Use the Angle Sum identity:

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

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

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

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 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}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{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}

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:

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}

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

Simplify

(- \frac{1}{2}) \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2} : \frac{2 ( - 1 - 3 )}{4}

(- \frac{1}{2}) \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

Remove parentheses:

(- a) = - a

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

Factor out common term

\frac{2}{2}

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

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

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

Apply rule

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 ( - 1 - 3 )}{4}

= \frac{2 ( - 1 - 3 )}{4}

sin (8 5^{\circ}) cos (2 5^{\circ}) - sin (2 5^{\circ}) cos (8 5^{\circ})

1 + 2 cos (π)

cos (2 \cdot \frac{3 π}{2})

9 sin (3 0^{\circ})

cos (- \frac{2}{5})

What is the value of cos(pi/4+(2pi)/3) ?
The value of cos(pi/4+(2pi)/3) is (sqrt(2)(-1-sqrt(3)))/4