Q:

NEED HELP HERE PLEASE6.051. Express the complex number in trigonometric form.-2 (1 point) 2(cos 90° + i sin 90°) 2(cos 0° + i sin 0°) 2(cos 180° + i sin 180°) 2(cos 270° + i sin 270°)2. Express the complex number in trigonometric form.-2i (1 point) 2(cos 180° + i sin 180°) 2(cos 90° + i sin 90°) 2(cos 270° + i sin 270°) 2(cos 0° + i sin 0°)3. Express the complex number in trigonometric form.6 - 6i (2 points) six square root two times the quantity cosine of seven pi divided by four plus i times sine of seven pi divided by four six square root two times the quantity cosine of five pi divided by four plus i times sine of five pi divided by four six times the quantity cosine of seven pi divided by four plus i times sine of seven pi divided by four six times the quantity cosine of five pi divided by four plus i times sine of five pi divided by four4. Write the complex number in the form a + bi.3(cos 270° + i sin 270°) (2 points) -3i 3i 3 -35. Find the product of z1 and z2, where z1 = 7(cos 40° + i sin 40°), and z2 = 6(cos 145° + i sin 145°). (2 points) 42(cos 40° + i sin 40°) 42(cos 185° + i sin 185°) 13(cos 185° + i sin 185°) 42(cos 5800° + i sin 5800°)6. Find the cube roots of 27(cos 330° + i sin 330°). (2 points)

Accepted Solution

A:
1. The answer is C. cos180=-1, sin180=0. 2*(cos180+i*sin180)=2*(-1+0)=-2. Check every other answer, none of which gets -2. 

2. The answer is C. cos270=0, sin270=-1. (You can draw out these angles to see). 2*(cos270+i*sin270)=2*(0-i)=-2i, as desired. Other choices don't work.

3. Answer A. Modulus of z is \sqrt(6^2+(-6)^2)=6*\sqrt(2). The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion. Theta=arctan(-6/6), and arctan(-1)=-pi/4, so theta=-pi/4=-pi/4+2pi=7pi/4. So A is the correct answer.

4. The answer is A. As above, cos270=0, sin270=-1. 3(cos270+sin270*i)=3*(0-i)=-3i. This problem is similar to question 2.

5. z1 = 7(cos 40° + i sin 40°), and z2 = 6(cos 145° + i sin 145°). z1*z2=7*6*(cos 40° + i sin 40°)*(cos 145° + i sin 145°)=42*(cos40*cos145-sin40*sin145+i*sin40*cos145+i*sin145*cos40). Use formula for sum/difference formula of cosines, cos40*cos145-sin40*sin145=cos(40+145)=cos185. Again, sin40*cos145+sin145*cos40=sin(40+145)=sin185. The answer is 42(cos 185° + i sin 185°).