if z= r ( cos tita + i.sin tita ) then the value of z/z + z/z is equal to
Answers
Answer:
Given : z = r( cosθ + i sin θ )
To Find : \dfrac{z}{\overline{z}} +\dfrac{\overline{z}}{z}
z
z
+
z
z
Solution:
z = r( cosθ + i sin θ )
\overline{z}
z
= r( cosθ - i sin θ )
\dfrac{z}{\overline{z}} +\dfrac{\overline{z}}{z}
z
z
+
z
z
= r( cosθ + i sin θ ) /r ( cosθ - i sin θ ) + r( cosθ - i sin θ ) /r( cosθ + i sin θ )
= ( cosθ + i sin θ ) / ( cosθ - i sin θ ) + ( cosθ - i sin θ ) / ( cosθ + i sin θ )
= ( ( cosθ + i sin θ )² + ( cosθ - i sin θ )²) /(cos²θ - i² sin² θ)
= ( 2( cos²θ + i² sin² θ )) /(cos²θ - i² sin² θ)
i² = - 1
= ( 2( cos²θ - sin² θ )) /(cos²θ + sin² θ)
cos²θ + sin² θ = 1
cos²θ - sin² θ = cos2θ
= 2 cos2θ
\dfrac{z}{\overline{z}} +\dfrac{\overline{z}}{z}
z
z
+
z
z
= 2 cos2θ
Step-by-step explanation:
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