Corrigés - inéquation trigonométrique - TS2

Corrigé 1

Dans $]-\pi\ ;\ \pi]$

1. $x\in\left\lbrace -\dfrac{3\pi}{4}\;,-\dfrac{5\pi}{12}\;,\dfrac{\pi}{4}\;,\dfrac{7\pi}{12}\right\rbrace$

2. $x\in\left\lbrace -\dfrac{2\pi}{3}\;,-\dfrac{\pi}{2}\;,0\;,\dfrac{\pi}{6}\;,\dfrac{2\pi}{3}\;,\dfrac{5\pi}{6}\right\rbrace$

3. $x\in\left\lbrace 0\;,\dfrac{\pi}{2}\right\rbrace$

4. $x\in\left\lbrace -\dfrac{\pi}{2}\;,-\dfrac{\pi}{4}\;,\dfrac{\pi}{2}\;,\dfrac{3\pi}{4}\right\rbrace$

5. $x\in\left\lbrace -\dfrac{\pi}{2}\;,0\;,\dfrac{\pi}{2}\right\rbrace$

6. $x\in\left\lbrace -\dfrac{23\pi}{36}\;,-\dfrac{11\pi}{36}\;,\dfrac{\pi}{36}\;,\dfrac{13\pi}{36}\;,\dfrac{25\pi}{36}\right\rbrace$

7. $x\in\left\lbrace -\dfrac{17\pi}{24}\;,-\dfrac{25\pi}{48}\;,-\dfrac{\pi}{48}\;,\dfrac{7\pi}{24}\;,\dfrac{23\pi}{48}\;,\dfrac{47\pi}{48}\right\rbrace$

8. $x\in \left\lbrace -\dfrac{37\pi}{60}\;,-\dfrac{13\pi}{60}\;,\dfrac{11\pi}{60}\;,\dfrac{5\pi}{12}\;,\dfrac{7\pi}{12}\;,\dfrac{59\pi}{60}\right\rbrace$

9. $x\in\left\lbrace -\dfrac{19\pi}{24}\;,-\dfrac{3\pi}{4}\;,-\dfrac{7\pi}{24}\;,\dfrac{5\pi}{24}\;,\dfrac{\pi}{4}\;,\dfrac{17\pi}{24}\right\rbrace$

10. $x\in \left\lbrace -\dfrac{59\pi}{72}\;,-\dfrac{35\pi}{72}\;,-\dfrac{17\pi}{24}\;,-\dfrac{11\pi}{72}\;,\dfrac{13\pi}{72}\;,\dfrac{37\pi}{72}\;,\dfrac{61\pi}{72}\right\rbrace$

11. $x\in\left\lbrace -\dfrac{\pi}{2}\;,\dfrac{\pi}{2}\right\rbrace$

12. $x\in \left\lbrace -\dfrac{5\pi}{12}\;,\dfrac{7\pi}{12}\right\rbrace$

Corrigé 2

Dans $]-\pi\ ;\ \pi]$

1. $x\in\left\lbrace -\dfrac{2\pi}{3}\;,-\dfrac{\pi}{3}\;,\dfrac{\pi}{3}\;,\dfrac{2\pi}{3}\right\rbrace$

2. $x\in\left\lbrace -\dfrac{\pi}{2}\;,-\dfrac{\pi}{6}\;,\dfrac{\pi}{2}\;,\dfrac{5\pi}{6}\right\rbrace$

3. $x=\left\lbrace -\dfrac{\pi}{24}+\dfrac{k\pi}{4}\;,k=-3\;,\ldots\;,4\right\rbrace$

4. $x\in\left\lbrace-\dfrac{2\pi}{3}\;,-\dfrac{\pi}{6}\;,\dfrac{\pi}{3}\;,\dfrac{5\pi}{6}\right\rbrace$

5. $x=\dfrac{\pi}{36}+\dfrac{k\pi}{6}\;,k=-6\;,\ldots\;,5$

6. $x\in\left\lbrace -\dfrac{17\pi}{24}\;,-\dfrac{13\pi}{24}\;,-\dfrac{5\pi}{24}\;,\dfrac{7\pi}{24}\;,\dfrac{11\pi}{24}\;,\dfrac{19\pi}{24}\;,\dfrac{23\pi}{24}\right\rbrace$

7. $x\in\left\lbrace -\dfrac{3\pi}{4}\;,-\dfrac{5\pi}{12}\;,-\dfrac{\pi}{12}\;,\dfrac{\pi}{4}\;,\dfrac{7\pi}{12}\;,\dfrac{11\pi}{12}\right\rbrace$

8. $x\in\left\lbrace -\dfrac{\pi}{3}\;,\dfrac{2\pi}{3}\right\rbrace$

Exercice 3

Dans $[0\ ;\ 2\pi[$

1. $\left(\cos x+1\right)\left(2\cos x+\sqrt{3}\right)=0\ :\ x\in\left\lbrace\dfrac{\pi}{6}\;,\dfrac{5\pi}{6}\;,\pi\;,\dfrac{7\pi}{6}\right\rbrace$

2. $4\cos^{2} x-3\cos x=0\ :\ x\in\left(\dfrac{\pi}{2}\;,\dfrac{3\pi}{2}\;,\arccos\dfrac{3}{4}\;,2\pi-\arccos\dfrac{3}{4}\right\rbrace$

3. $\cos 2x\left(2\cos 2x+1\right)=0\ :\ x\in\left\lbrace \dfrac{\pi}{4}\;,\dfrac{\pi}{3}\;,\dfrac{2\pi}{3}\;,\dfrac{3\pi}{4}\;,\dfrac{5\pi}{4}\;,\dfrac{4\pi}{3}\;,\dfrac{5\pi}{3}\;,\dfrac{7\pi}{4}\right\rbrace$

4. $2\cos x\left(1+\cos 4x\right)=0\ :\ x\in\left\lbrace \dfrac{\pi}{4}\;,\dfrac{\pi}{2}\;,\dfrac{3\pi}{4}\;,\dfrac{5\pi}{4}\;,\dfrac{3\pi}{2}\;,\dfrac{7\pi}{4}\right\rbrace$

5. $\cos 3x\left(2\cos 2x-1\right)=0\ :\ x\in\left\lbrace \dfrac{\pi}{6}\;,\dfrac{\pi}{2}\;,\dfrac{5\pi}{6}\;,\dfrac{7\pi}{6}\;,\dfrac{3\pi}{2}\;,\dfrac{11\pi}{6}\right\rbrace$

6. Discriminant =$-32+8\sqrt{3}< 0$ : aucune solution réelle.

7. $\sin 2x\left(2\cos x+1\right)=0\ :\ x\in\left\lbrace 0\;,\dfrac{\pi}{2}\;,\dfrac{2\pi}{3}\;,\pi\;,\dfrac{4\pi}{3}\;,\dfrac{3\pi}{2}\right\rbrace$

8. $2\sin 3x\cos x=0\ :\ x\in\left\lbrace 0\;,\dfrac{\pi}{3}\;,\dfrac{\pi}{2}\;,\dfrac{2\pi}{3}\;,\pi\;,\dfrac{4\pi}{3}\;,\dfrac{3\pi}{2}\;,\dfrac{5\pi}{3}\right\rbrace$

9. $1+\dfrac{2}{\sqrt{3}}\tan x-\tan^{2} x=0\Longleftrightarrow \tan x=\sqrt{3}\text{ ou }-\dfrac{1}{\sqrt{3}}\ :\ x\in\left\lbrace \dfrac{\pi}{3}\;,\dfrac{5\pi}{6}\;,\dfrac{4\pi}{3}\;,\dfrac{11\pi}{6}\right\rbrace$

10. $\left(\tan x+1\right)\left(\sqrt{3}\tan x+1\right)=0\ :\ x\in\left\lbrace \dfrac{3\pi}{4}\;,\dfrac{5\pi}{6}\;,\dfrac{7\pi}{4}\;,\dfrac{11\pi}{6}\right\rbrace$

11. $\left(\tan x-1\right)\left(\tan x+\sqrt{3}\right)=0\ :\ x\in\left\lbrace \dfrac{\pi}{4}\;,\dfrac{2\pi}{3}\;,\dfrac{5\pi}{4}\;,\dfrac{5\pi}{3}\right\rbrace$

(La question $5^{\circ}$ apparaît en double dans l'énoncé imprimé ; seule la première version, ci-dessus, a été corrigée.)

Corrigé 4

1. $\sin x=\dfrac{4}{5}\;,\tan x=-\dfrac{4}{3}$

2. $\sin x=-\dfrac{2}{3}\;,\tan x=-\dfrac{2\sqrt{5}}{5}$

3. $\sin x=\dfrac{5}{13}\;,\tan x=-\dfrac{5}{12}$

Corrigé 5

1. $\sin x=\dfrac{2\sqrt{5}}{5}\;,\cos x=-\dfrac{\sqrt{5}}{5}$

2. $\sin x=-\dfrac{11}{61}\;,\cos x=-\dfrac{11}{61}\;,\cos x=-\dfrac{60}{61}$

3. $\sin x=-\dfrac{8}{17}\;,\cos x=\dfrac{15}{17}$

Corrigé 6

1. $\cos x=-\dfrac{\sqrt{3}}{3}\;,\tan x=-\sqrt{2}$

2. $\cos x=-\dfrac{3}{5}\;,\tan x=\dfrac{4}{3}$

3. $\cos x=\dfrac{\sqrt{7}}{3}\;,\tan x=-\sqrt{\dfrac{2}{7}}$

Corrigé 7 

$\tan 2x=\dfrac{2\left(\sqrt{3}-2\right)}{1-\left(\sqrt{3}-2\right)^{2}}=-\dfrac{\sqrt{3}}{3}$

comme $2x\in]-\pi\;,0[\;,2x=-\dfrac{\pi}{6}$ donc $x=-\dfrac{\pi}{12}$

Corrigé 8

$\sin x=\dfrac{ \sqrt{2}+\sqrt{6}}{4}=\sin\left(\dfrac{5\pi}{12}\right)$, avec $x\in\left] 0\;,\dfrac{\pi}{2}\right[$

Donc $x=\dfrac{5\pi}{12}$ et $\cos 2x=-\dfrac{\sqrt{3}}{2}$