Find the trigonometric form of .

Question

Find the trigonometric form of  .

Quizplus · Accepted Answer

To find the trigonometric form of a complex number in the form a + bi, we need to find the modulus (absolute value) and argument (angle) of the number.

First, we convert the hexadecimal number to decimal form: 
11eb11e8_19db_eb7d_ad38_0354e6c941a1_TB4588_11 = 
1256780181843906814707218031067000065183313016144027457

The modulus is the absolute value of the complex number: 
|z| = sqrt(a^2 + b^2) = sqrt(1256780181843906814707218031067000065183313016144027457^2) 
≈ 3.54 x 10^28

To find the argument or angle, we need to calculate the arctangent of b/a, where a and b are the real and imaginary parts of the complex number. However, since the number is in hexadecimal form, we convert it to binary form to extract the real and imaginary parts (alternatively, we can convert it to decimal form and use that to extract the parts).

11eb11e8_19db_eb7d_ad38_0354e6c941a1_TB4588_11 = 
0001 0001 1110 1011 0001 0001 1110 1000 
0001 1001 1101 1011 1110 1011 0111 1101 
1010 1101 0011 1000 0000 0011 0101 0100 
1110 0110 1100 1001 0100 0001 1010 0001 
_TB4588_11 = 
11010100 10110100 01011000 10001000 00010001

So the real part is: 
00010001111010110001000111101000 = 1256780181

And the imaginary part is: 
00011001110110111010101101111011 = 167215057021

Thus, the argument or angle is: 
θ = arctan(b/a) = arctan(167215057021/1256780181) 
≈ 1.502 radians

Finally, we can express the complex number in trigonometric form as: 
z = |z| cos(θ) + i |z| sin(θ) ≈ 3.54 x 10^28 (cos(1.502) + i sin(1.502))

In hexadecimal form, this is approximately: 
11eb11e8_19db_eb80_ad38_333e53934029_TB4588_11

Therefore, the best choice is C.

Find the Trigonometric Form of