Trigonometry Complex Number &Polar Form?

Question

1)Evaluate the expression i^112 and write the result in the a+bi form .

The real number a equals_____

The real number b equals _____


2)If the Polar Form expression for a complex number is

9[cos(6pi/21)+ (i)sin(6pi/21)] then the rectangular form is a+bi, where a is _________ and b is ___________

Answer 1

Only one problem per question please.

1) Evaluate the expression i^112 and write the result in the a+bi form.

Please note the pattern of i^n

i
i² = -1
i³ = -1
i^4 = 1

And the cycle is completed every fourth power. After dividing the power by 4, only the remainder matters. So:

i^112 = (i^4)^28 = 1^28 = 1

This has no remainder (a remainder of zero).

Answer 2

think of of the complicated quantity like a vector. The horizontal axis (frequently the x - axis) is for the genuine area. The vertical axis is for the imaginary area. The quantity you have been given might translate right into a vector beginning on the beginning and going sqrt 3 interior the effective x course and one million interior the damaging y course. this implies that is interior the fourth quadrant. to locate the perspective, take the inverse tangent of the imaginary coefficient divided by using the genuine area (as a effect, that would desire to be sqrt 3 divided by using damaging one million). once you place this into your calculator, it supply you the effect of -30 levels, that's precise, yet you're in many situations predicted to apply a great perspective. as a effect, you may desire to locate the coterminal perspective, by using including 360 levels, which grants 330. to locate the quantity in front of the parentheses, you may desire to locate the scale of that vector by using squaring the two areas, including them, and taking the sq. root of the sum. sqrt (3) squared is 3. damaging one million squared is one million. 3 + one million = 4. The sq. root of four is two. So the polar form is two(cos 330 + i sin 330)