Exercice 1 On sait que cos 2 x + sin 2 x = 1 \cos^2 x+\sin^2 x=1 cos 2 x + sin 2 x = 1 x x x cos 2 x = 1 − sin 2 x \cos^2 x=1-\sin^2 x cos 2 x = 1 − sin 2 x cos 2 x = 1 − ( 1 3 ) 2 = 8 9 \cos^2 x=1-\left(\dfrac{1}{3}\right)^2=\dfrac{8}{9} cos 2 x = 1 − ( 3 1 ) 2 = 9 8 cos x = 2 2 3 \cos x=\dfrac{2\sqrt{2}}{3} cos x = 3 2 2 cos x = − 2 2 3 \cos x=-\dfrac{2\sqrt{2}}{3} cos x = − 3 2 2
Sachant que x ∈ [ π 2 ; π ] x\in\left[\dfrac{\pi}{2};\pi\right] x ∈ [ 2 π ; π ] − 1 ≤ cos x ≤ 0 -1\leq\cos x\leq 0 − 1 ≤ cos x ≤ 0 cos x = − 2 2 3 \cos x=-\dfrac{2\sqrt{2}}{3} cos x = − 3 2 2 tan x = sin x cos x = − 1 2 2 = − 2 4 \tan x=\dfrac{\sin x}{\cos x}=-\dfrac{1}{2\sqrt{2}}=-\dfrac{\sqrt{2}}{4} tan x = cos x sin x = − 2 2 1 = − 4 2
Exercice 2 On commence par déterminer la mesure principale de l’angle, c’est-à-dire la mesure comprise dans ] − π ; π ] ]-\pi;\pi] ] − π ; π ]
65 π 4 = 8 × 8 π + π 4 = 8 × 2 π + π 4 \dfrac{65\pi}{4}=\dfrac{8\times8\pi+\pi}{4}=8\times2\pi+\dfrac{\pi}{4} 4 65 π = 4 8 × 8 π + π = 8 × 2 π + 4 π
π 4 \dfrac{\pi}{4} 4 π 65 π 4 \dfrac{65\pi}{4} 4 65 π
Comme pour tout entier relatif k k k cos ( x + 2 k π ) = cos ( x ) \cos(x+2k\pi)=\cos(x) cos ( x + 2 k π ) = cos ( x )
On obtient : cos ( 65 π 4 ) = cos ( π 4 ) = 2 2 \cos\left(\dfrac{65\pi}{4}\right)=\cos\left(\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2} cos ( 4 65 π ) = cos ( 4 π ) = 2 2
Procédons de même.
− 39 π 4 = − 40 π 4 + π 4 = − 10 π + π 4 = − 5 × 2 π + π 4 -\dfrac{39\pi}{4}=-\dfrac{40\pi}{4}+\dfrac{\pi}{4}=-10\pi+\dfrac{\pi}{4}=-5\times2\pi+\dfrac{\pi}{4} − 4 39 π = − 4 40 π + 4 π = − 10 π + 4 π = − 5 × 2 π + 4 π
π 4 \dfrac{\pi}{4} 4 π − 39 π 4 -\dfrac{39\pi}{4} − 4 39 π
Par conséquent : sin ( − 39 π 4 ) = sin ( π 4 ) = 2 2 \sin\left(-\dfrac{39\pi}{4}\right)=\sin\left(\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2} sin ( − 4 39 π ) = sin ( 4 π ) = 2 2
Exercice 3 cos ( − x ) = cos ( x ) \cos(-x)=\cos(x) cos ( − x ) = cos ( x ) cos ( x + π / 2 ) = − sin ( x ) \cos(x+\pi/2)=-\sin(x) cos ( x + π /2 ) = − sin ( x ) cos ( x + π ) = − cos ( x ) \cos(x+\pi)=-\cos(x) cos ( x + π ) = − cos ( x ) cos ( x + 2 π ) = cos ( x ) \cos(x+2\pi)=\cos(x) cos ( x + 2 π ) = cos ( x ) cos ( π − x ) = − cos ( x ) \cos(\pi-x)=-\cos(x) cos ( π − x ) = − cos ( x ) cos ( π / 2 − x ) = sin ( x ) \cos(\pi/2-x)=\sin(x) cos ( π /2 − x ) = sin ( x )
Calculons sin ( π 8 ) \sin\left(\dfrac{\pi}{8}\right) sin ( 8 π ) sin 2 ( π 8 ) = 2 − 2 4 \sin^2\left(\dfrac{\pi}{8}\right)=\dfrac{2-\sqrt{2}}{4} sin 2 ( 8 π ) = 4 2 − 2 sin ( π 8 ) > 0 \sin\left(\dfrac{\pi}{8}\right)>0 sin ( 8 π ) > 0 sin ( π 8 ) = 1 2 2 − 2 \sin\left(\dfrac{\pi}{8}\right)=\dfrac{1}{2}\sqrt{2-\sqrt{2}} sin ( 8 π ) = 2 1 2 − 2
cos ( − π 8 ) = cos ( π 8 ) = 1 2 2 + 2 \cos\left(-\dfrac{\pi}{8}\right)=\cos\left(\dfrac{\pi}{8}\right)=\dfrac{1}{2}\sqrt{2+\sqrt{2}} cos ( − 8 π ) = cos ( 8 π ) = 2 1 2 + 2
cos ( 3 π 8 ) = cos ( 4 π − π 8 ) = cos ( π 2 − π 8 ) = sin ( π 8 ) = 1 2 2 − 2 \cos\left(\dfrac{3\pi}{8}\right)=\cos\left(\dfrac{4\pi-\pi}{8}\right)=\cos\left(\dfrac{\pi}{2}-\dfrac{\pi}{8}\right)=\sin\left(\dfrac{\pi}{8}\right)=\dfrac{1}{2}\sqrt{2-\sqrt{2}} cos ( 8 3 π ) = cos ( 8 4 π − π ) = cos ( 2 π − 8 π ) = sin ( 8 π ) = 2 1 2 − 2
cos ( 5 π 8 ) = cos ( π 2 + π 8 ) = − sin ( π 8 ) = − 1 2 2 − 2 \cos\left(\dfrac{5\pi}{8}\right)=\cos\left(\dfrac{\pi}{2}+\dfrac{\pi}{8}\right)=-\sin\left(\dfrac{\pi}{8}\right)=-\dfrac{1}{2}\sqrt{2-\sqrt{2}} cos ( 8 5 π ) = cos ( 2 π + 8 π ) = − sin ( 8 π ) = − 2 1 2 − 2
cos ( 9 π 8 ) = cos ( π + π 8 ) = − cos ( π 8 ) = − 1 2 2 + 2 \cos\left(\dfrac{9\pi}{8}\right)=\cos\left(\pi+\dfrac{\pi}{8}\right)=-\cos\left(\dfrac{\pi}{8}\right)=-\dfrac{1}{2}\sqrt{2+\sqrt{2}} cos ( 8 9 π ) = cos ( π + 8 π ) = − cos ( 8 π ) = − 2 1 2 + 2
cos ( − 325 π 8 ) = cos ( 325 π 8 ) \cos\left(\dfrac{-325\pi}{8}\right)=\cos\left(\dfrac{325\pi}{8}\right) cos ( 8 − 325 π ) = cos ( 8 325 π ) cos ( 325 π 8 ) = cos ( 20 × ( 2 π ) + 5 π 8 ) = cos ( 5 π 8 ) = − 1 2 2 − 2 \cos\left(\dfrac{325\pi}{8}\right)=\cos\left(20\times(2\pi)+\dfrac{5\pi}{8}\right)=\cos\left(\dfrac{5\pi}{8}\right)=-\dfrac{1}{2}\sqrt{2-\sqrt{2}} cos ( 8 325 π ) = cos ( 20 × ( 2 π ) + 8 5 π ) = cos ( 8 5 π ) = − 2 1 2 − 2