Algebra 2 / Trig Honors Polar Coordinates NotesPolar-1: Polar Coordinates:

(r, a)r = radius, angle = degrees/radians

If (–3, 45) then start at –3 (to the left) then rotate 45° from there

Answers to all the graphs except #9, 12 & 13 are off one click

Polar-2:

14. x2 – 4x + 4 + y2 = 4 x2 + y2 = 4x + 0 r2 = 4(rcos𝛳) r = 4cos𝛳17. tan–1 y

x = π

3

yx=tan (60 °)

Polar 3a & b: Graphing (do both on the same day)

Polar-4: Complex Polars

Given a complex number z = a + bi     the modulus of a complex number is denoted by  |z| (radius) = √a2+b2

complex number in Rectangular form to Polar form a + bi → r(cos 𝛳 + i sin 𝛳) graph a + bi find the radius and angle

1. 3 + 3i |z| = √32+32 = 3√2 tan 𝛳 = ba so = tan–1 33 so = 45°

Polar form: 3√2 (cos 45 + i sin 45)

Short-hand forr(cos 𝛳 + i sin 𝛳) =r cis 𝛳

7. 3 cis π/4 = 3(cos 45 + i sin 45) 3( √2

2 + i √2

2 ) = 3√2

2+3√2

2i Rectangular form

11. 2 cis 3 (radians) = 2(cos 3 (171.9°) + i sin 3)

Polar-5: https://www.youtube.com/watch?v=CiJnYud_KMo

Algebra 2 / Trig Honors Polar Coordinates Notes

De Moivre's Theorem: Any two complex numbers in polar form,   and   are

multiplied as such:

1, 2(3) cis (150 + 300) 6 cis 450 = 6(cos 90 + i sin 90) 6(0 + 1i) = 0 + 6i

And divided:

Answers to 11, 12, 16 are wrong, should be …

11. −√33

−13i 12, 0 + √6

2i 16. 0 –

13i

Polar-6: De Moivre's Theorem exponentiated:

Convert each complex number into polar form (r cis θ ).

1. 1 + 1i so r= √12+12 = √2 & 𝛳 = 45°

so (√220) cis 20(45)

1024 cos 900 + 1024 sin 900

–1024 + 0i

Primary root: In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

4th root of … n = 14

7. (r cis θ )14 so r=√64+192 = 16 𝛳 = 120°

161/4 cis 30 = 2(√32 )+2( 1

2i) = √3+i

Polar-7: nth roots of a number

In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P.

Algebra 2 / Trig Honors Polar Coordinates Notes

It follows from this (and the fundamental theorem of algebra), that if the degree of a real polynomial is odd, it must have at least one real root. That fact can also be proven by using the intermediate value theorem.

The polynomial x2 + 1 = 0 has roots ±i. Any real square matrix of odd degree has at least one real eigenvalue. For example, if the

matrix is orthogonal, then 1 or −1 is an eigenvalue.

The polynomial

has roots

and thus can be factored as

In computing the product of the last two factors, the imaginary parts cancel, and we get

The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors.

Complex conjugate, (or just the conjugate).  Given the complex number z = a + bi  the complex conjugate is denoted by z  and is defined to be, z=a−bi

3 cube roots of –8 → x3 = –8 → x3 + 8 = 0 (x+ 2)(x2 – 2x + 4) = 0

x + 2 = 0 or x2 – 2x + 4 = 0

x = –2 or x=−(−2)±√(−2)2−4 (1)(4)2(1)

¿ 2±√−122 ¿ 2± i2√3

2

{1+i √3 ,−2 ,1−i√3

– 8 + 0 i= 8 cis 180

1

2

3

Algebra 2 / Trig Honors Polar Coordinates Notes

Finding all the nth roots of a complex number

1. From Polar-6 2 cis (30 + 0) 2 cis (30 + 90) 2 cis (30 + 180) 2 cis (30 + 270)

1

2

34