Argument relationships and tangent identities

A question is this type if and only if it uses arguments of complex numbers to prove trigonometric identities involving arctan, typically by considering arg(z₁z₂) = arg(z₁) + arg(z₂).

8 questions

CAIE P3 2014 November Q5

5 The complex numbers $w$ and $z$ are defined by $w = 5 + 3 \mathrm { i }$ and $z = 4 + \mathrm { i }$.

Express $\frac { \mathrm { i } w } { z }$ in the form $x + \mathrm { i } y$, showing all your working and giving the exact values of $x$ and $y$.
Find $w z$ and hence, by considering arguments, show that $$\tan ^ { - 1 } \left( \frac { 3 } { 5 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right) = \frac { 1 } { 4 } \pi$$

CAIE P3 2022 June Q5

5 The complex number $3 - \mathrm { i }$ is denoted by $u$.

Show, on an Argand diagram with origin $O$, the points $A , B$ and $C$ representing the complex numbers $u , u ^ { * }$ and $u ^ { * } - u$ respectively. State the type of quadrilateral formed by the points $O , A , B$ and $C$.
Express $\frac { u ^ { * } } { u }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
By considering the argument of $\frac { u ^ { * } } { u }$, or otherwise, prove that $\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$.

CAIE P3 2020 November Q6

6 The complex number $u$ is defined by $$u = \frac { 7 + \mathrm { i } } { 1 - \mathrm { i } }$$

Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
Show on a sketch of an Argand diagram the points $A , B$ and $C$ representing $u , 7 + \mathrm { i }$ and $1 - \mathrm { i }$ respectively.
By considering the arguments of $7 + \mathrm { i }$ and $1 - \mathrm { i }$, show that $$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = \tan ^ { - 1 } \left( \frac { 1 } { 7 } \right) + \frac { 1 } { 4 } \pi$$

OCR MEI Paper 3 2021 November Q12

12 Show that $\beta = \arctan \left( \frac { 1 } { 3 } \right)$, as given in line 15 .

OCR MEI Paper 3 2021 November Q13

Use triangle ABE in Fig. C 2 to show that $\arctan x + \arctan \left( \frac { 1 } { x } \right) = \frac { \pi } { 2 }$, as given in line 29 .
Sketch the graph of $\mathrm { y } = \arctan \mathrm { x }$.
What property of the arctan function ensures that $\mathrm { y } > \frac { 1 } { \mathrm { x } } \Rightarrow \arctan y > \arctan \left( \frac { 1 } { \mathrm { x } } \right)$, as given in line 30 ?

OCR MEI Paper 3 Specimen Q14

14 Show that the two values of $b$ given on line 36 are equivalent.

Edexcel CP2 2021 June Q9

(a) Given that $| z | < 1$, write down the sum of the infinite series

$$1 + z + z ^ { 2 } + z ^ { 3 } + \ldots$$ (b) Given that $z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )$,

use the answer to part (a), and de Moivre's theorem or otherwise, to prove that $$\frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$
show that the sum of the infinite series $1 + z + z ^ { 2 } + z ^ { 3 } + \ldots$ cannot be purely imaginary, giving a reason for your answer.

CAIE P3 2012 June Q7

Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
Show on a sketch of an Argand diagram the points $A , B$ and $C$ representing the complex numbers $u , 1 + 2 \mathrm { i }$ and $1 - 3 \mathrm { i }$ respectively.
By considering the arguments of $1 + 2 \mathrm { i }$ and $1 - 3 \mathrm { i }$, show that $$\tan ^ { - 1 } 2 + \tan ^ { - 1 } 3 = \frac { 3 } { 4 } \pi$$