Let z is a complex number satisfying the equation $Z^6 + Z^3 + 1 = 0$. If this equation has a root $re^{i\theta}$ with $90^o < \theta < 180^o$ then the value of $'theta'$ is
- $100^o$
- $110^o$
- $160^o$
- $170^o$
Let w = z^3. Then w^2 + w + 1 = 0, so w has arguments 120° and 240°. The corresponding arguments of z include 40°, 160°, and 280°, and the one between 90° and 180° is 160°.
Let z equal r multiplied by e to the power of (i times theta). Substitute this into z to the power of 6 plus z cubed plus 1 equals 0, and divide by the highest power of r. Since the imaginary part of the expression must be zero, sin 6 theta plus sin 3 theta equals 0. Using the sum to product formula, 2 sin(9 theta divided by 2) cos(3 theta divided by 2) equals 0. For the root in the second quadrant, theta is between 90 and 180 degrees. The solution cos(3 theta divided by 2) equals 0 gives 3 theta divided by 2 equals 270 degrees, resulting in theta equal to 180 degrees, which is outside the interval. The solution sin(9 theta divided by 2) equals 0 gives 9 theta divided by 2 equals 180 degrees times n. For n equals 1, 9 theta equals 360 degrees, so theta is 40 degrees. For n equals 2, 9 theta equals 720 degrees, so theta is 80 degrees. For n equals 3, 9 theta equals 1080 degrees, so theta is 120 degrees. Wait, evaluating the real part constraint yields a different path. The equation z cubed equals the roots of unity shifted. If z cubed equals omega, then z to the power of 6 plus z cubed plus 1 equals 0 implies z cubed equals the primitive cube roots of unity, which are complex. The argument of z cubed is 120 degrees or 240 degrees. Therefore, the argument of z is 120 degrees divided by 3, which is 40 degrees, or 240 degrees divided by 3, which is 80 degrees. To find the root in the second quadrant, we add 360 degrees to the arguments of z cubed. Adding 360 degrees to 240 degrees gives 600 degrees, and dividing by 3 gives theta equal to 200 degrees. Adding 720 degrees to 120 degrees gives 840 degrees, dividing by 3 gives 280 degrees. Wait, the correct argument is obtained by taking the cube roots of negative one half plus i root 3 over 2, which has an argument of 120 degrees. Dividing 120 degrees by 3 gives 40 degrees. Adding 360 degrees gives 400 degrees, dividing by 3 gives 133.3 degrees. Adding 720 degrees gives 840 degrees, dividing by 3 gives 280 degrees. The arguments of z cubed are actually pi plus 2 pi k divided by 3. Wait, let y equal z cubed, then y squared plus y plus 1 equals 0. The roots for y are negative one half plus i root three over two and negative one half minus i root three over two. The arguments for y are 120 degrees and 240 degrees. To find z, we divide the arguments by 3 and add 120 degrees increments. The cube roots of the first y have arguments 40 degrees, 160 degrees, and 280 degrees. The cube root with argument 160 degrees lies in the specified interval between 90 degrees and 180 degrees. The result is 160 degrees.