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

The value of cos((2pi)/3+(3pi)/4) is (sqrt(2)-sqrt(6))/4

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

cos (\frac{2 π}{3} + \frac{3 π}{4})

\frac{2 - 6}{4}

Solution steps

cos (\frac{2 π}{3} + \frac{3 π}{4})

Rewrite using trig identities:

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

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

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{2}

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

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

Simplify

(- \frac{2}{2}) (- \frac{1}{2}) - \frac{2}{2} \cdot \frac{3}{2} : \frac{2 - 6}{4}

= \frac{2 - 6}{4}

cos (arctan (0))

10 \cdot cos (5 0^{\circ})

6 (cos (\frac{5 π}{6}) + i sin (\frac{5 π}{6}))

sin (- 1 (0))

cos (- 51 0^{\circ})

What is the value of cos((2pi)/3+(3pi)/4) ?
The value of cos((2pi)/3+(3pi)/4) is (sqrt(2)-sqrt(6))/4