grandes-ecoles 2010 QI.A.1

grandes-ecoles · France · centrale-maths1__pc Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation

For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
a) Show that the functions $F_n$ are defined on the same domain $D$ which should be specified.
b) Calculate $F_1(x), F_2(x)$ and $F_3(x)$ for all $x \in D$.
c) Calculate $F_n(1), F_n(0)$ and $F_n(-1)$ for all $n \in \mathbb{N}$.
d) Specify the parity properties of $F_n$ as a function of $n$.

For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.

a) Show that the functions $F_n$ are defined on the same domain $D$ which should be specified.

b) Calculate $F_1(x), F_2(x)$ and $F_3(x)$ for all $x \in D$.

c) Calculate $F_n(1), F_n(0)$ and $F_n(-1)$ for all $n \in \mathbb{N}$.

d) Specify the parity properties of $F_n$ as a function of $n$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.C QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2