Exercise 13.4 Archives

Exercise 13.4

Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 – sin α + i cos α
(ii) (1 – i) / (cos π/3 + i sin π/3)

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: (i) $1-\sin \alpha+i \cos \alpha$ Given that $Z=1-\sin \alpha+i \cos a$ Using the formulas, $\operatorname{Sin}^{2} \theta+\cos ^{2} \theta=1$ $\operatorname{Sin} 2 \theta=2 \sin \theta...

Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 + i tan α
(ii) tan α – i

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: (i) $1+\mathrm{i} \tan \alpha$ Given that $Z=1+\mathrm{i}$ tan $\alpha$ It is known to us that the polar form of a complex number $Z=$ In which, $\begin{array}{l} \left.|Z|=\text { modulus...

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) 1 – i
(ii) (1 – i) / (1 + i)

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: It is known to us that the polar form of a complex number $Z=x+i$ iy is given by $Z=|Z|(\cos \theta+i \sin \theta)$ In which, $\begin{array}{l} \left.|Z|=\text { modulus of complex number...

If $\mathbf{z}_{1}$ and $\mathbf{z}_{2}$ are two complex number such that $\left|\mathbf{z}_{1}\right|=\left|\mathbf{z}_{2}\right|$ and $\arg \left(\mathbf{z}_{1}\right)+\arg \left(\mathbf{z}_{2}\right)=\pi$, then show that $\mathbf{z}_{1}=-\overline{\mathbf{z}_{2}}$

CBSE, Class 11, Complex Numbers, Exercise 13.4, Maths, RD Sharma

Solution: Given that $\left|z_{1}\right|=\left|z_{2}\right|$ and $\arg \left(z_{1}\right)+\arg \left(z_{2}\right)=\pi$ Let's assume $\arg \left(z_{1}\right)=\theta$ $\arg...

Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 – sin α + i cos α
(ii) (1 – i) / (cos π/3 + i sin π/3)

Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 + i tan α
(ii) tan α – i

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) 1 – i
(ii) (1 – i) / (1 + i)

If $\mathbf{z}_{1}$ and $\mathbf{z}_{2}$ are two complex number such that $\left|\mathbf{z}_{1}\right|=\left|\mathbf{z}_{2}\right|$ and $\arg \left(\mathbf{z}_{1}\right)+\arg \left(\mathbf{z}_{2}\right)=\pi$, then show that $\mathbf{z}_{1}=-\overline{\mathbf{z}_{2}}$

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Express the following complex numbers in the form r (cos θ + i sin θ): (i) 1 – sin α + i cos α (ii) (1 – i) / (cos π/3 + i sin π/3)

Express the following complex numbers in the form r (cos θ + i sin θ): (i) 1 + i tan α (ii) tan α – i

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form: (i) 1 – i (ii) (1 – i) / (1 + i)

If $\mathbf{z}_{1}$ and $\mathbf{z}_{2}$ are two complex number such that $\left|\mathbf{z}_{1}\right|=\left|\mathbf{z}_{2}\right|$ and $\arg \left(\mathbf{z}_{1}\right)+\arg \left(\mathbf{z}_{2}\right)=\pi$, then show that $\mathbf{z}_{1}=-\overline{\mathbf{z}_{2}}$

Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 – sin α + i cos α
(ii) (1 – i) / (cos π/3 + i sin π/3)

Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 + i tan α
(ii) tan α – i

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) 1 – i
(ii) (1 – i) / (1 + i)