To find: the different polar forms for a given complex number.

The polar forms of $3−3i$ are

$32cos7π4+isin7π4$ or $32cos7π4+2nπ+isin7π4+2nπ$ for integer $n$ .

Given information:

Given a complex number in standard form is $3−3i$ .

Formula used:

Polar form of the complex number:

The polar form of the complex number $z=a+bi$ is $z=rcosθ+isinθ,$ $a=rcosθ,$ $b=rsinθ,$ $r=a2+b2,$ and $tanθ=ba$ .

If $θ$ is the argument of a complex number then for any integer $n$ , $θ+2nπ$ is also an argument of $θ$ .

Calculation:

Modulus of a complex number:

Substitute the values $3$ for $a$ , $−3$ for $b$ in $r=a2+b2$ to get the modulus $r$ of $3−3i$ :

$r=a2+b2=32+−32=9+9=18=32$

Thus, $r=32$ .

Argument of complex number:

Substitute the values $3$ for $a$ , $−3$ for $b$ in $tanθ=ba$ to get argument $θ$ of $3−3i$ :

$tanθ=batanθ=−33tanθ=−1θ=tan−1−1=−π4$

Given $0≤θ≤2π$ , and reference angle $θ′$ for $θ$ is $−π/4$ . Then the required angle is:

$θ=2π+−π4=7π4$

Thus, the arguments $θ$ of $3−3i$ are $7π4$ and $7π4+2nπ$ , for integer $n$ .

Polar form of complex number:

Substitute $32$ for $r$ , $7π4$ for $θ$ in $rcosθ+isinθ$ to get polar form of $3−3i$ :

$rcosθ+isinθ=32cos7π4+isin7π4$

Thus, the polar forms of $3−3i$ are

$32cos7π4+isin7π4$

Or

$32cos7π4+2nπ+isin7π4+2nπ$ for integer $n$ .

Chapter 6, Problem 38RE

Chapter 6, Problem 40RE

Chapter 6 Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

Ch. 6.1 - Prob. 1QR Ch. 6.1 - Prob. 2QR Ch. 6.1 - Prob. 3QR Ch. 6.1 - Prob. 4QR Ch. 6.1 - Prob. 5QR Ch. 6.1 - Prob. 6QR Ch. 6.1 - Prob. 7QR Ch. 6.1 - Prob. 8QR Ch. 6.1 - Prob. 9QR Ch. 6.1 - Prob. 10QR

Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - Prob. 13E Ch. 6.1 - Prob. 14E Ch. 6.1 - Prob. 15E Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - 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