What is the value of cos((3pi)/4)+i*sin((3pi)/4) ?

The value of cos((3pi)/4)+i*sin((3pi)/4) is -(sqrt(2))/2+i(sqrt(2))/2

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cos((3pi)/4)+i*sin((3pi)/4)

cos (\frac{3 π}{4}) + i \cdot sin (\frac{3 π}{4})

- \frac{2}{2} + i \frac{2}{2}

Solution steps

cos (\frac{3 π}{4}) + i sin (\frac{3 π}{4})

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 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{3 π}{4}) = \frac{2}{2}

sin (\frac{3 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{2}{2} + i \frac{2}{2}

cos^{3} (9 0^{\circ})

4 + \frac{3 2}{2} cos (\frac{3 π}{4})

\frac{1}{cos ^{2} ( \frac{π}{3} )}

arccos (\frac{5}{9 14})

cos (- 5 0^{\circ})

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