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

The value of sqrt(2)(cos(pi/4)+isin(pi/4)) is 1+i

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sqrt(2)(cos(pi/4)+isin(pi/4))

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

1 + i

Solution steps

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

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{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}

= 2 (\frac{2}{2} + i \frac{2}{2})

Simplify

2 (\frac{2}{2} + i \frac{2}{2}) : 1 + i

2 (\frac{2}{2} + i \frac{2}{2})

Multiply

i \frac{2}{2} : \frac{2 i}{2}

i \frac{2}{2}

Multiply fractions:

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

= \frac{2 i}{2}

= 2 (\frac{2 i}{2} + \frac{2}{2})

Combine the fractions

\frac{2}{2} + \frac{2 i}{2} : \frac{2 + 2 i}{2}

Apply rule

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

= \frac{2 + 2 i}{2}

= 2 (\frac{2 + 2 i}{2})

Remove parentheses:

(a) = a

= 2 \frac{2 + 2 i}{2}

Multiply fractions:

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

= \frac{( 2 + 2 i ) 2}{2}

Cancel

\frac{( 2 + 2 i ) 2}{2} : \frac{2 + 2 i}{2}

\frac{( 2 + 2 i ) 2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} ( 2 + 2 i )}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{2 + 2 i}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{2 + 2 i}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{2 + 2 i}{2}

= \frac{2 + 2 i}{2}

Factor out common term

2

= \frac{2 ( 1 + i )}{2}

Cancel the common factor:

2

= 1 + i

= 1 + i

sec (- \frac{9 π}{4})

- 2 cos (\frac{5 π}{6})

sec (\frac{- 2 π}{3})

8 cos (4 0^{\circ})

7 cos (5 0^{\circ})

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