Trigonometriska ekvationer

Trigonometriskekv_f



































\\L\ddot{o}sning: \\\\a)\: \: sin\, x=0,53 \\sin^{-1}\, 0,53\approx 32.0^{\circ},\, vilket\, ger\, oss\, tv\aa \, fall: \\\\\left\{\begin{matrix} x\approx 32^{\circ}+n\cdot 360^{\circ}\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ x\approx 180^{\circ}-32^{\circ}+n\cdot 360^{\circ}=148^{\circ}+n\cdot 360^{\circ} \end{matrix}\right.

\\\\\\b)\: cos(2x+15^{\circ})=\frac{1}{\sqrt{2}} \\\\cos^{-1}\frac{1}{\sqrt{2}}=45,0^{\circ},\, vilket\, ger\, oss\, de\, tv\aa \, fallen: \\\\\left\{\begin{matrix} 2x+15=45,0^{\circ}+n\cdot 360^{\circ}\: \: \\ 2x+15=-45,0^{\circ}+n\cdot 360^{\circ} \end{matrix}\right. \\\\\left\{\begin{matrix} 2x=45,0^{\circ}-15^{\circ}+n\cdot 360^{\circ}=30^{\circ}+n\cdot 360^{\circ}\: \: \: \: \: \: \: \: \: \: \: \\ 2x+15=-45,0^{\circ}-15^{\circ}+n\cdot 360^{\circ}=-60^{\circ}+n\cdot 360^{\circ} \end{matrix}\right. \\\\\left\{\begin{matrix} x=15^{\circ}+n\cdot 180^{\circ}\: \: \\ x=-30^{\circ}+n\cdot 180^{\circ} \end{matrix}\right. \\\\\\c)\: tan\, 2x=\sqrt{3} \\\\tan^{-1}\, \sqrt{3}=60^{\circ} \\2x=60^{\circ}+n\cdot 180^{\circ} \\x=30^{\circ}+n\cdot 90^{\circ}