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Rappel N° 9 - Trigonométrie⚓︎

Valeurs remarquables de cosinus et sinus⚓︎

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Recherche 9-1

Donner les valeurs :

  1. \(\cos \left(\dfrac{3 \pi}{4}\right) \)

  2. \(\tan \left(\dfrac{5 \pi}{4}\right) \)

  3. \(\cos \left(\dfrac{\pi}{6}\right) \)

  4. \(\tan \left(\dfrac{\pi}{4}\right)\)

  5. \(\cos (7 \pi)\)

  6. \(\sin \left(\dfrac{7 \pi}{6}\right) \)

sol

Recherche 9-2

Simplifier :

  1. \(\cos \left(\dfrac{\pi}{4}\right)+\cos \left(\dfrac{3 \pi}{4}\right)+\cos \left(\dfrac{5 \pi}{4}\right)+\cos \left(\dfrac{7 \pi}{4}\right) \)

  2. \(\cos ^{2}\left(\dfrac{4 \pi}{3}\right)+\sin ^{2}\left(\dfrac{4 \pi}{3}\right)\)

  3. \(\sin \left(\dfrac{5 \pi}{6}\right)+\sin \left(\dfrac{7 \pi}{6}\right)\)

  4. \(\cos ^{2}\left(\dfrac{4 \pi}{3}\right)-\sin ^{2}\left(\dfrac{4 \pi}{3}\right)\)

  5. \(\tan \left(\dfrac{2 \pi}{3}\right)+\tan \left(\dfrac{3 \pi}{4}\right)+\tan \left(\dfrac{5 \pi}{6}\right)+\tan \left(\dfrac{7 \pi}{6}\right) \)

sol

Signe du cosinus et du sinus⚓︎

Recherche 9-3

Donner le signe :

  1. \(\cos \left(\dfrac{2 \pi}{5}\right) \)

  2. \(\cos \left(\dfrac{8 \pi}{5}\right) \)

  3. \(\sin \left(\dfrac{14 \pi}{5}\right) \)

  4. \(\sin \left(\dfrac{7 \pi}{5}\right) \)

  5. \(\tan \left(\dfrac{13 \pi}{5}\right) \)

  6. \(\tan \left(-\dfrac{3 \pi}{5}\right) \)

sol

Propriétés remarquables de cosinus et sinus⚓︎

Recherche 9-4

Simplifier :

  1. \(\sin (\pi-x)+\cos \left(\dfrac{\pi}{2}+x\right)\)
  2. \(\sin \left(\dfrac{\pi}{2}-x\right)+\sin \left(\dfrac{\pi}{2}+x\right)\)
  3. \(\sin (-x)+\cos (\pi+x)+\sin \left(\dfrac{\pi}{2}-x\right)\)
  4. \(\cos (x-\pi)+\sin \left(-\dfrac{\pi}{2}-x\right)\)

sol

Formules de duplication⚓︎

On rappelle les formules suivantes : \(\sin (2 x)=2 \sin (x) \cos (x) \quad\) et \(\quad \cos (2 x)=\cos ^{2}(x)-\sin ^{2}(x)\)

Recherche 9-5

En remarquant qu'on a \(\dfrac{\pi}{4}=2 \times \dfrac{\pi}{8}\), calculer :

  1. \(\cos \left(\dfrac{\pi}{8}\right)\)
  2. \(\sin \left(\dfrac{\pi}{8}\right)\)
  3. \(\tan \left(\dfrac{\pi}{8}\right)\)
  1. On a \(\cos \left(\frac{\pi}{4}\right)=2 \cos ^{2}\left(\frac{\pi}{8}\right)-1\) donc \(\cos ^{2}\left(\frac{\pi}{8}\right)=\frac{\frac{\sqrt{2}}{2}+1}{2}=\frac{\sqrt{2}+2}{4}\).

    De plus, \(\cos \left(\frac{\pi}{8}\right) \geqslant 0\) donc \(\cos \left(\frac{\pi}{8}\right)=\frac{\sqrt{2+\sqrt{2}}}{2}\).

  2. On a \(\sin ^{2}\left(\frac{\pi}{8}\right)=1-\cos ^{2}\left(\frac{\pi}{8}\right)=\frac{2-\sqrt{2}}{4}\) et \(\sin \left(\frac{\pi}{8}\right) \geqslant 0\) donc \(\sin \left(\frac{\pi}{8}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}\).

  3. \(\tan \left(\frac{\pi}{8}\right)=\sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}}=\sqrt{\frac{(2-\sqrt{2})^{2}}{4-2}}=\frac{2-\sqrt{2}}{\sqrt{2}}=\sqrt{2}-1\)

Recherche 9-6

Simplifier pour \(x \in] 0, \dfrac{\pi}{2}[\),

  1. \(\dfrac{1-\cos (2 x)}{\sin (2 x)}\)
  2. \(\dfrac{\cos (2 x)}{\cos (x)}-\dfrac{\sin (2 x)}{\sin (x)}\)
  1. On a \(\cos (2 x)=1-2 \sin ^{2}(x)\) donc \(\frac{1-\cos (2 x)}{\sin (2 x)}=\frac{2 \sin ^{2}(x)}{2 \sin x \cos (x)}=\tan (x)\).
  2. \(\frac{\cos (2 x)}{\cos (x)}-\frac{\sin (2 x)}{\sin (x)}=\frac{\cos ^{2}(x)-\sin ^{2}(x)}{\cos (x)}-\frac{2 \sin (x) \cos (x)}{\sin (x)}=\frac{\cos ^{2}(x)-\sin ^{2}(x)}{\cos (x)}-\frac{2 \cos ^{2}(x)}{\cos (x)}=-\frac{\cos ^{2}(x)+\sin ^{2}(x)}{\cos (x)}\)

Équations et inéquations trigonométriques⚓︎

Recherche 9-7

Résoudre dans \([0,2 \pi]\), dans \([-\pi, \pi]\), puis dans \(\mathbb{R}\) les équations suivantes :

  1. \(\cos (x)=\dfrac{1}{2}\)
  2. \(\tan (x)=-1\)
  3. \(\sin (x)=-\dfrac{\sqrt{3}}{2}\)
  4. \(\cos ^{2}(x)=\dfrac{1}{2}\)
  5. \(\sin (x)=\cos \left(\dfrac{2 \pi}{3}\right)\)
  6. \(|\tan (x)|=\dfrac{1}{\sqrt{3}}\)
  1. RAS
  2. RAS
  3. RAS
  4. RAS
  5. Cela revient à résoudre « \(\cos (x)=\frac{\sqrt{2}}{2}\) ou \(\cos (x)=-\frac{\sqrt{2}}{2}\) ».

Recherche 9-8

Résoudre dans \([0,2 \pi]\), puis dans \([-\pi, \pi]\), les inéquations suivantes :

  1. \(\cos (x) \geqslant-\dfrac{\sqrt{2}}{2}\)
  2. \(|\sin (x)| \leqslant \dfrac{1}{2}\)
  3. \(\cos (x) \leqslant \cos \left(\dfrac{\pi}{3}\right)\)
  4. \(\tan (x) \geqslant 1\)
  5. \(\sin (x) \leqslant \dfrac{1}{2}\)
  6. \(\cos \left(x-\dfrac{\pi}{4}\right) \geqslant 0 \)
  1. RAS
  2. RAS
  3. RAS
  4. Cela revient à résoudre \(-\frac{1}{2} \leqslant \sin (x) \leqslant \frac{1}{2}\).
  5. RAS
  6. Si \(x \in[0,2 \pi]\), alors \(t=x-\frac{\pi}{4} \in\left[-\frac{\pi}{4}, 2 \pi-\frac{\pi}{4}\right]\). On résout donc \(\cos (t) \geqslant 0\) pour \(t \in\left[-\frac{\pi}{4}, 2 \pi-\frac{\pi}{4}\right]\) ce qui donne \(t \in\left[-\frac{\pi}{4}, \frac{\pi}{2}\right] \cup\left[\frac{3 \pi}{2}, \frac{7 \pi}{4}\right]\) et donc \(x \in\left[0, \frac{3 \pi}{4}\right] \cup\left[\frac{7 \pi}{4}, 2 \pi\right]\).

Réponses mélangées⚓︎

\[{\small \begin{aligned} & {\left[-\dfrac{3 \pi}{4},-\dfrac{\pi}{2}\left[\cup \left[\dfrac{\pi}{4}, \dfrac{\pi}{2}\left[\quad\left[-\pi, \dfrac{\pi}{6}\right] \cup\left[\dfrac{5 \pi}{6}, \pi\right] \quad\left\{-\dfrac{5 \pi}{6},-\dfrac{\pi}{6}\right\} \quad\left\{-\dfrac{5 \pi}{6},-\dfrac{\pi}{6}, \dfrac{\pi}{6}, \dfrac{5 \pi}{6}\right\}\right.\right.\right.\right.} \\ & \dfrac{\sqrt{2+\sqrt{2}}}{2} \quad\left\{\dfrac{\pi}{4}+k \pi, k \in \mathbb{Z}\right\} \quad 1 \quad \tan x \quad \dfrac{\sqrt{3}}{2} \quad-\dfrac{1}{\cos (x)} \quad-\dfrac{1}{2} \\ & {\left[-\dfrac{\pi}{4}, \dfrac{3 \pi}{4}\right] \quad\left[0, \dfrac{\pi}{6}\right] \cup\left[\dfrac{5 \pi}{6}, 2 \pi\right] \quad\left\{\dfrac{\pi}{4}, \dfrac{5 \pi}{4}\right\} \quad\left\{\dfrac{\pi}{3}+2 k \pi, k \in \mathbb{Z}\right\} \cup\left\{-\dfrac{\pi}{3}+2 k \pi, k \in \mathbb{Z}\right\}} \\ & 0 \quad\left\{\dfrac{4 \pi}{3}+2 k \pi, k \in \mathbb{Z}\right\} \cup\left\{\dfrac{5 \pi}{3}+2 k \pi, k \in \mathbb{Z}\right\} \quad\left\{-\dfrac{3 \pi}{4}, \dfrac{\pi}{4}\right\} \quad\left\{\dfrac{\pi}{4}, \dfrac{3 \pi}{4}, \dfrac{5 \pi}{4} \dfrac{7 \pi}{4}\right\} \\ & \left\{\dfrac{\pi}{6}+k \pi, k \in \mathbb{Z}\right\} \cup\left\{\dfrac{5 \pi}{6}+k \pi, k \in \mathbb{Z}\right\} \quad\left[-\dfrac{3 \pi}{4}, \dfrac{3 \pi}{4}\right] \quad\left\{\dfrac{\pi}{4}+k \dfrac{\pi}{2}, k \in \mathbb{Z}\right\} \quad<0 \\ & {\left[0, \dfrac{3 \pi}{4}\right] \cup\left[\dfrac{7 \pi}{4}, 2 \pi\right] \quad\left[-\pi,-\dfrac{\pi}{3}\right] \cup\left[\dfrac{\pi}{3}, \pi\right] \quad\left[-\pi,-\dfrac{5 \pi}{6}\right] \cup\left[-\dfrac{\pi}{6}, \dfrac{\pi}{6}\right] \cup\left[\dfrac{5 \pi}{6}, \pi\right]>0} \\ & 2 \cos x \quad-2 \cos x \quad\left\{-\dfrac{\pi}{3}, \dfrac{\pi}{3}\right\} \quad \dfrac{\sqrt{2-\sqrt{2}}}{2} \quad\left[\dfrac{\pi}{3}, \dfrac{5 \pi}{3}\right] \quad\left\{\dfrac{\pi}{6}, \dfrac{5 \pi}{6}, \dfrac{7 \pi}{6}, \dfrac{11 \pi}{6}\right\} \\ & -\dfrac{1}{2} \quad\left\{\dfrac{-2 \pi}{3}, \dfrac{-\pi}{3}\right\} \quad\left[\dfrac{\pi}{4}, \dfrac{\pi}{2}\left[\cup \left[\dfrac{5 \pi}{4}, \dfrac{3 \pi}{2}\left[\begin{array}{llll} 0 & -1 & 0 & 1 \end{array}\right.\right.\right.\right. \\ & \left\{-\dfrac{3 \pi}{4},-\dfrac{\pi}{4}, \dfrac{\pi}{4}, \dfrac{3 \pi}{4}\right\} \quad\left\{\dfrac{7 \pi}{6}+2 k \pi, k \in \mathbb{Z}\right\} \cup\left\{\dfrac{11 \pi}{6}+2 k \pi, k \in \mathbb{Z}\right\} \quad\left\{\dfrac{7 \pi}{6}, \dfrac{11 \pi}{6}\right\} \\ & \sqrt{2}-1 \quad\left[0, \dfrac{3 \pi}{4}\right] \cup\left[\dfrac{5 \pi}{4}, 2 \pi\right] \quad\left\{\dfrac{\pi}{3}, \dfrac{5 \pi}{3}\right\} \quad\left\{\dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}\right\} \quad>0 \quad<0 \quad 1 \\ & -\dfrac{\sqrt{2}}{2}-\sin x \quad\left[0, \dfrac{\pi}{6}\right] \cup\left[\dfrac{5 \pi}{6}, \dfrac{7 \pi}{6}\right] \cup\left[\dfrac{11 \pi}{6}, 2 \pi\right] \quad>0 \quad-1-\sqrt{3} \end{aligned}} \]