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Question

Mathematics Question on Complex Numbers and Quadratic Equations

Amplitude of the complex number i sin⁡(π19)i\, \sin \bigg( \frac{\pi}{19} \bigg) is

A

π19 \frac{\pi}{19}

B

−π19- \frac{\pi}{19}

C

π2 \frac{\pi}{2}

D

π2−π19 \frac{\pi}{2} - \frac{\pi}{19}

Answer

π2 \frac{\pi}{2}

Explanation

Solution

Let z=i sin⁡π19z = i \, \sin \frac{\pi}{19}
or z=0+i sin⁡π19z = 0 + i \, \sin \frac{\pi}{19}
Let amplitude of zz be θ\theta then
sin⁡θ=sin⁡π1902+(sin⁡π19)2[sin⁡θ=ba2+b2]\sin \theta =\frac{\sin \frac{\pi}{19}}{\sqrt{0^{2}+\left(\sin \frac{\pi }{19}\right)^{2}}} \left[\sin \theta =\frac{b}{\sqrt{a^{2}+b^{2}}}\right]
or sin⁡θ=sin⁡π19sin⁡π19=1⇒θ=π2\sin\theta =\frac{\sin \frac{\pi }{19}}{\sin \frac{\pi }{19}} =1 \Rightarrow \theta =\frac{\pi}{2}