Algebra

Polar Coordinates

Polar Coordinates - Multiplication

         

Let z=3(cos(15)+isin(15)),z = 3(\cos(15 ^\circ) + i\sin(15 ^\circ)), w=5(cos(54)+isin(54)).w = 5(\cos(54 ^\circ)+i\sin(54 ^\circ)). Then zwzw can be expressed as r(cosα+isinβ)r(\cos \alpha ^ \circ + i\sin\beta ^ \circ ), where rr is a real number, 0α90 0 \leq \alpha \leq 90 and 0β900 \leq \beta \leq 90. What is r+α+βr+\alpha+\beta?

Let z1z_1 and z2z_2 be complex numbers such that z1=15(cos512π+isin512π),z_1 = 15 \left( \cos \frac{5}{12}\pi + i \sin \frac{5}{12}\pi \right), z2=2(cos112π+isin112π).z_2 = 2 \left( \cos \frac{1}{12}\pi + i \sin \frac{1}{12}\pi \right). The product z1z2z_1 z_2 can be expressed as a+bi, a + bi, where aa and bb are real numbers. What is the value of a+b?a+b?

Details and assumptions

ii is the imaginary number satisfying i2=1i^2 = -1.

Consider the complex numbers z1=5+53i,z2=3(cosπ6+isinπ6).\begin{aligned} z_1 &= 5+5\sqrt{3}i, \\ z_2 &= 3\left(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}\right). \end{aligned} If the product z1z2z_1z_2 can be expressed as a+bia+bi, where aa and bb are real numbers, what is a+ba+b?

Let z1,z2z_1, z_2 and z3z_3 be complex numbers such that z1=2(cos112π+isin112π),z2=73(cos112π+isin112π),z3=9(cos16π+isin16π).\begin{aligned} z_1 &= 2 \left( \cos \frac{1}{12}\pi + i \sin \frac{1}{12}\pi \right), \\ z_2 &= 7 \sqrt{3} \left( \cos \frac{1}{12}\pi + i \sin \frac{1}{12}\pi \right), \\ z_3 &= 9 \left( \cos \frac{1}{6}\pi + i \sin \frac{1}{6}\pi \right). \end{aligned} If the product z1z2z3z_1 z_2 z_3 can be expressed as a+bi, a + bi, where aa and bb are real numbers, what is b2a2 \frac{b^2}{a^2} ?

Let z1z_1 and z2z_2 be complex numbers such that z1=10(cos124π+isin124π),z_1 = 10 \left( \cos \frac{1}{24}\pi + i \sin \frac{1}{24}\pi \right), z2=52(cos524π+isin524π).z_2 = 5 \sqrt{2} \left( \cos \frac{5}{24}\pi + i \sin \frac{5}{24}\pi \right). The product z1z2z_1 z_2 can be expressed as a+bi, a + bi, where aa and bb are real numbers. What is the value of a+b a+b ?

Details and assumptions

ii is the imaginary number that satisfies i2=1i^2 = -1.

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