FP2.3 - Further complex numbers

Euler’s relation eiθ = cos θ + i sin θ.; Students should be familiar with cos θ = (eiθ + e−iθ) and sin θ = (eiθ − e−iθ). 2i.

Use fp2.3.1 - euler’s relation eiθ = cos θ + i sin θ to connect the rule to the data and decision in the question.

This matters because fp2.3.1 - euler’s relation eiθ = cos θ + i sin θ determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.3.1 - euler’s relation eiθ = cos θ + i sin θ to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.3.1 - Euler’s relation eiθ = cos θ + i sin θ is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

De Moivre’s theorem and its To include finding cos nθ and sin mθ in terms of powers of application to trigonometric sin θ and cos θ and also powers of sin θ and cos θ in terms identities and to roots of a complex of multiple angles.; Students should be able to prove number.; De Moivre’s theorem for any integer n.

Use fp2.3.2 - de moivre’s theorem and its to connect the rule to the data and decision in the question.

This matters because fp2.3.2 - de moivre’s theorem and its determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.3.2 - de moivre’s theorem and its to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.3.2 - De Moivre’s theorem and its is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Interpret and sketch complex-number loci and regions defined by modulus and argument conditions, including |z − a| = b, |z − a| = k|z − b| and arg(z − a) = β.

Use fp2.3.3 - loci and regions in the argand diagram to connect the rule to the data and decision in the question.

This matters because fp2.3.3 - loci and regions in the argand diagram determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.3.3 - loci and regions in the argand diagram to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.3.3 - Loci and regions in the Argand diagram is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Elementary transformations from az + b Transformations such as w = z2 and w =, where the z-plane to the w-plane. cz + d a, b, c, d ∈, may be set. ℂ.

Use fp2.3.4 - transformations between the z-plane and w-plane to connect the rule to the data and decision in the question.

This matters because fp2.3.4 - transformations between the z-plane and w-plane determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.3.4 - transformations between the z-plane and w-plane to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.3.4 - Transformations between the z-plane and w-plane is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.