Let$A = 2$cos$\$theta+i$sin$$\theta {$$\theta$$\i_n$[0, 2$\pi$]:$1 + 10$Re( ) = 0}. cos$\$theta-3i$sin$$\theta Then$$\sum$$\theta$$\inA$$\theta 2 is equal to$
Step-by-Step Solution
Key Concept: Simplify the real part of the complex fraction and solve the resulting trigonometric equation on$[0,2\pi]$.
$1 + 10$Re( 2 cos$\$theta + i$sin$$\theta ) = 0$(1)$cos$$\$theta - 3i$sin$$\theta$¯ ¯¯ ∴$z + z = 2$Re(z) 2 cos$\$theta + i$sin$$\theta 2 cos$$\$theta - i$sin$$\$theta -1$+ = 2$\times ( ) cos$\$theta - 3i$sin$$\theta cos$$\$theta + 3i$sin$$\theta 10 2 2 2 2 (2 cos$$\$theta - 3$sin$$\theta) + (2 cos$$\theta) - (3 sin$$\theta) -$2 = 2$cos 2$$\$theta + 9$sin$$\theta 10 2 2 2 cos$$\$theta - 3$sin$$\theta -1$$\Rightarrow$ = 2 2 10 cos$\$theta + 9$sin$$\theta 2 2 2 2$\Rightarrow 20 cos$\$theta - 30$sin$$\theta = - cos$$\$theta - 9$sin$$\theta 2 2 21 cos$$\$theta - 21$sin$$\theta = 0$$\Rightarrow$ cos 2$\theta = 0$$\pi 3$$\pi 5$$\pi 7$$\pi 2$$\theta =,,, 2 2 2 2 2 2 2 2 2 2$$\pi 9$$\pi 25$$\pi 49$$\pi 84$$\pi 21$$\pi 2$$\Rightarrow$$\sum$$\theta = + + + = = 16 16 16 16 16 4$
Correct Answer: 1