Solve root i - ScanMath

Question

Simplify the expression

cos (\frac{π}{4}) + i sin (\frac{π}{4}) cos (\frac{5 π}{4}) + i sin (\frac{5 π}{4})

Alternative Form

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

Evaluate

i

Rewrite the complex number in polar form

More Steps

1 \times (cos (\frac{π}{2}) + i sin (\frac{π}{2}))

Calculate the nth roots of a complex r (cos (θ) + i \times sin (θ),using n z = n r (cos \frac{θ + 2 kπ}{n} + i sin \frac{θ + 2 kπ}{n})

1 \times (cos (\frac{\frac{π}{2} + 2 kπ}{2}) + i sin (\frac{\frac{π}{2} + 2 kπ}{2}))

Simplify

1 \times (cos (\frac{\frac{π}{2} + 2 kπ}{2}) + i sin (\frac{\frac{π}{2} + 2 kπ}{2}))

Simplify

cos (\frac{\frac{π}{2} + 2 kπ}{2}) + i sin (\frac{\frac{π}{2} + 2 kπ}{2})

Since n = 2,substitute k = 0, 1 into the expression

cos (\frac{\frac{π}{2} + 2 \times 0 \times π}{2}) + i sin (\frac{\frac{π}{2} + 2 \times 0 \times π}{2}) cos (\frac{\frac{π}{2} + 2 \times 1 \times π}{2}) + i sin (\frac{\frac{π}{2} + 2 \times 1 \times π}{2})

Calculate

More Steps

cos (\frac{π}{4}) + i sin (\frac{π}{4}) cos (\frac{\frac{π}{2} + 2 \times 1 \times π}{2}) + i sin (\frac{\frac{π}{2} + 2 \times 1 \times π}{2})

Solution

More Steps

cos (\frac{π}{4}) + i sin (\frac{π}{4}) cos (\frac{5 π}{4}) + i sin (\frac{5 π}{4})

Alternative Form

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

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