1 Une mesure d'un angle géométrique est 250 grades. Donner sa mesure en degré et en radian. \( \square \) Repréenter sur un cercle trigonométrique les points \( M_{k}\left(-\frac{\pi}{6}+k \pi\right) \) tel que ( \( k \in Z \) ) 3) Calculer \( \cos \left(\frac{-79 \pi}{3}\right) \) et \( \tan \left(\frac{41 \pi}{4}\right) \). 1) Simplifier les expressions suivantes (en indiquant les propriétis utilisers) : - \( A=\cos (x-38 \pi)+\sin (251 \pi-x)+3 \cos \left(\frac{17 \pi}{2}+x\right)+2 \sin \left(\frac{15 \pi}{2}-x\right) \) - \( B=\cos (x+34 \pi)+\sin (57 \pi-x)+3 \cos \left(\frac{13 \pi}{2}-x\right)+2 \sin \left(\frac{3 \pi}{2}-x\right) \) 5 5. On pose, \( \sin \left(\frac{\pi}{5}\right)=a \). Calculer en fonction de \( a \) les rapports trigonométriques suivants : (a) \( \cos \left(\frac{\pi}{5}\right) \) et \( \tan \left(\frac{\pi}{5}\right) \) (D) \( \cos \left(\frac{6 \pi}{5}\right) \) et \( \tan \left(\frac{11 \pi}{10}\right) \)

Simplify the expression by following steps: - step0: Solution: \(-\frac{a}{\sqrt{1-a^{2}}}\) - step1: Simplify: \(-\frac{a\sqrt{1-a^{2}}}{1-a^{2}}\) Calculate or simplify the expression \( \sqrt(1-a^2) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{1-a^{2}}\) Calculate or simplify the expression \( a/\sqrt(1-a^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{a}{\sqrt{1-a^{2}}}\) - step1: Simplify: \(\frac{a\sqrt{1-a^{2}}}{1-a^{2}}\) Calculate or simplify the expression \( \cos(x-38*\pi)+\sin(251*\pi-x)+3*\cos(17*\pi/2+x)+2*\sin(15*\pi/2-x) \). Simplify the expression by following steps: - step0: Solution: \(\cos\left(x-38\pi \right)+\sin\left(251\pi -x\right)+3\cos\left(\frac{17\pi }{2}+x\right)+2\sin\left(\frac{15\pi }{2}-x\right)\) - step1: Rewrite the expression: \(\cos\left(x-38\pi \right)+\sin\left(251\pi -x\right)+3\left(-\sin\left(x\right)\right)+2\sin\left(\frac{15\pi }{2}-x\right)\) - step2: Rewrite the expression: \(\cos\left(x-38\pi \right)+\sin\left(251\pi -x\right)+3\left(-\sin\left(x\right)\right)+2\left(-\cos\left(x\right)\right)\) - step3: Rewrite the expression: \(\cos\left(x\right)+\sin\left(251\pi -x\right)+3\left(-\sin\left(x\right)\right)+2\left(-\cos\left(x\right)\right)\) - step4: Rewrite the expression: \(\cos\left(x\right)+\sin\left(x\right)+3\left(-\sin\left(x\right)\right)+2\left(-\cos\left(x\right)\right)\) - step5: Rewrite the expression: \(\cos\left(x\right)+\sin\left(x\right)-3\sin\left(x\right)+2\left(-\cos\left(x\right)\right)\) - step6: Rewrite the expression: \(\cos\left(x\right)+\sin\left(x\right)-3\sin\left(x\right)-2\cos\left(x\right)\) - step7: Subtract the terms: \(-\cos\left(x\right)-2\sin\left(x\right)\) Calculate or simplify the expression \( \cos(-79*\pi/3) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(\frac{-79\pi }{3}\right)\) - step1: Rewrite the fraction: \(\cos\left(-\frac{79\pi }{3}\right)\) - step2: Rewrite the expression: \(\cos\left(\frac{79\pi }{3}\right)\) - step3: Rearrange the numbers: \(\cos\left(\frac{\pi }{3}+13\times 2\pi \right)\) - step4: Rearrange the terms: \(\cos\left(\frac{\pi }{3}\right)\) - step5: Calculate the trigonometric value: \(\frac{1}{2}\) Calculate or simplify the expression \( -\sqrt(1-a^2) \). Simplify the expression by following steps: - step0: Solution: \(-\sqrt{1-a^{2}}\) Calculate or simplify the expression \( \tan(41*\pi/4) \). Calculate the value by following steps: - step0: Calculate: \(\tan\left(\frac{41\pi }{4}\right)\) - step1: Rearrange the numbers: \(\tan\left(\frac{\pi }{4}+10\pi \right)\) - step2: Rearrange the terms: \(\tan\left(\frac{\pi }{4}\right)\) - step3: Calculate the trigonometric value: \(1\) Calculate or simplify the expression \( \cos(x+34*\pi)+\sin(57*\pi-x)+3*\cos(13*\pi/2-x)+2*\sin(3*\pi/2-x) \). Simplify the expression by following steps: - step0: Solution: \(\cos\left(x+34\pi \right)+\sin\left(57\pi -x\right)+3\cos\left(\frac{13\pi }{2}-x\right)+2\sin\left(\frac{3\pi }{2}-x\right)\) - step1: Rewrite the expression: \(\cos\left(x+34\pi \right)+\sin\left(57\pi -x\right)+3\sin\left(x\right)+2\sin\left(\frac{3\pi }{2}-x\right)\) - step2: Rewrite the expression: \(\cos\left(x+34\pi \right)+\sin\left(57\pi -x\right)+3\sin\left(x\right)+2\left(-\cos\left(x\right)\right)\) - step3: Rewrite the expression: \(\cos\left(x\right)+\sin\left(57\pi -x\right)+3\sin\left(x\right)+2\left(-\cos\left(x\right)\right)\) - step4: Rewrite the expression: \(\cos\left(x\right)+\sin\left(x\right)+3\sin\left(x\right)+2\left(-\cos\left(x\right)\right)\) - step5: Rewrite the expression: \(\cos\left(x\right)+\sin\left(x\right)+3\sin\left(x\right)-2\cos\left(x\right)\) - step6: Subtract the terms: \(-\cos\left(x\right)+4\sin\left(x\right)\) 1. **Conversion de 250 grades en degrés et en radians :** - Pour convertir 250 grades en degrés, on utilise la relation \( 1 \text{ grade} = \frac{9}{10} \text{ degrés} \). - Donc, \( 250 \text{ grades} = 250 \times \frac{9}{10} = 225 \text{ degrés} \). - Pour convertir en radians, on utilise la relation \( 180 \text{ degrés} = \pi \text{ radians} \). - Ainsi, \( 225 \text{ degrés} = 225 \times \frac{\pi}{180} = \frac{5\pi}{4} \text{ radians} \). 2. **Représentation sur un cercle trigonométrique des points \( M_{k}\left(-\frac{\pi}{6}+k \pi\right) \) :** - Les points \( M_k \) représentent des angles qui se répètent tous les \( \pi \) radians. - Pour \( k \in \mathbb{Z} \), cela signifie que nous avons des angles de la forme \( -\frac{\pi}{6} \), \( \frac{5\pi}{6} \), \( \frac{11\pi}{6} \), etc., qui se trouvent sur le cercle trigonométrique. 3. **Calcul de \( \cos \left(\frac{-79 \pi}{3}\right) \) :** - Le résultat est \( \frac{1}{2} \). 4. **Calcul de \( \tan \left(\frac{41 \pi}{4}\right) \) :** - Le résultat est \( 1 \). 5. **Simplification des expressions :** - Pour \( A = \cos (x-38 \pi) + \sin (251 \pi - x) + 3 \cos \left(\frac{17 \pi}{2} + x\right) + 2 \sin \left(\frac{15 \pi}{2} - x\right) \): - En utilisant les propriétés des fonctions trigonométriques, on obtient \( A = -\cos(x) - 2\sin(x) \). - Pour \( B = \cos (x + 34 \pi) + \sin (57 \pi - x) + 3 \cos \left(\frac{13 \pi}{2} - x\right) + 2 \sin \left(\frac{3 \pi}{2} - x\right) \): - On obtient \( B = -\cos(x) + 4\sin(x) \). 6. **Calcul des rapports trigonométriques en fonction de \( a = \sin \left(\frac{\pi}{5}\right) \) :** - \( \cos \left(\frac{\pi}{5}\right) = \sqrt{1-a^{2}} \) - \( \tan \left(\frac{\pi}{5}\right) = \frac{a}{\sqrt{1-a^{2}}} \) - \( \cos \left(\frac{6 \pi}{5}\right) = -\sqrt{1-a^{2}} \) - \( \tan \left(\frac{11 \pi}{10}\right) = -\frac{a}{\sqrt{1-a^{2}}} \) Ces résultats fournissent une vue d'ensemble des conversions et simplifications demandées.