الحل العام للمعادلة المثلثية

1. ‫1‬
‫‪www.mathtop4.alafdal.net‬‬
‫ﻣﻨﺘﺪى اﻟﺮﯾﺎﺿﯿﺎت اﻟﻤﺘﻘﺪﻣﺔ‬
‫اﻟﺤﻞ اﻟﻌﺎم ﻟﻠﻤﻌﺎدﻟﺔ اﻟﻤﺜﻠﺜﯿﺔ‬
‫اوﻻ:اﻟﻤﻌﺎدﻟﺔ اﻟﺘﻰ ﻋﻠﻰ اﻟﺼﻮرة ﺟﺎ ‪ = α‬ﺟﺎ‪β‬‬
‫اﻟﺤﻞ اﻟﻌﺎم ﯾﻜﻮن ‪ = α‬ﻥﺐ + )-1( ﻥ ‪β‬‬
‫اﻟﺒﺮھﺎن‬
‫ﺉ ‪٢ = α‬ﻥﺐ + ‪β‬‬
‫اﻣﺎ‬
‫ﺟﺎ ‪ = α‬ﺟﺎ‪β‬‬
‫او‬
‫ﺟﺎ ‪ = α‬ﺟﺎ)ﺐ - ‪(β‬‬
‫ﺉ ‪ = α‬ﻋﺪد زوﺟﻰ×ﺐ + ‪β‬‬
‫ﺉ ‪٢ = α‬ﻥﺐ + ﺐ - ‪β‬‬
‫ﺉ ‪٢) = α‬ﻥ+1(ﺐ - ‪β‬‬
‫ﺉ ‪ = α‬ﻋﺪد ﻓﺮدى×ﺐ - ‪β‬‬
‫ﺇ اﻟﺤﻞ اﻟﻌﺎم ﯾﻜﻮن ‪ = α‬ﻥﺐ + )-1( ﻥ ‪β‬‬
‫ﻣﺜﺎل ﺗﻄﺒﯿﻘﻰ‬
‫أوﺟﺪ اﻟﺤﻞ اﻟﻌﺎم ﻟﻠﻤﻌﺎدﻟﺔ: ﺟﺎ٢‪ = θ‬ﺟﺘﺎ ‪θ‬‬
‫اﻟﺤﻞ‬
‫ﺟﺎ٢‪ = θ‬ﺟﺎ) ﺐ - ‪(θ‬‬
‫ﺐ‬
‫٢ ‪ = θ‬ﻥﺐ + )-1( ﻥ ٢) - ‪(θ‬‬
‫٢‬
‫ﺐ‬
‫)‬
‫٢‬
‫ ‪(θ‬‬‫اذا ﻛﺎﻧﺖ ن زوﺟﻰ ﻓﺎن ٢ ‪ = θ‬ﻥﺐ +‬
‫ﺐ‬
‫ﺐ‬
‫ﻥﺐ‬
‫+‬
‫ﺉ ‪=θ‬‬
‫٣‪ = θ‬ﻥﺐ +‬
‫٢‬
‫٦‬
‫٣‬
‫ﺐ‬
‫اذا ﻛﺎﻧﺖ ن ﻓﺮدى ﻓﺎن ٢ ‪ = θ‬ﻥﺐ - ) - ‪(θ‬‬
‫٢‪ = θ‬ﻥﺐ -‬
‫اﻋﺪاد اﻷﺳﺘﺎذ / اﺑﺮاھﯿﻢ اﻷﺣﻤﺪى اﺑﺮاھﯿﻢ‬
‫ﺐ‬
‫٢‬
‫٢‬
‫+‪θ‬‬
‫ﺉ ‪ = θ‬ﻥﺐ -‬
‫ﻣﻌﻠﻢ ﺧﺒﯿﺮ اﻟﺮﯾﺎﺿﯿﺎت‬
‫ﺐ‬
‫٢‬
‫2110 390 2110‬

2. ‫2‬
‫‪www.mathtop4.alafdal.net‬‬
‫ﺛﺎﻧﯿﺎ:اﻟﻤﻌﺎدﻟﺔ اﻟﺘﻰ ﻋﻠﻰ اﻟﺼﻮرة ﺟﺘﺎ ‪ = α‬ﺟﺘﺎ‪β‬‬
‫ﻣﻨﺘﺪى اﻟﺮﯾﺎﺿﯿﺎت اﻟﻤﺘﻘﺪﻣﺔ‬
‫اﻟﺤﻞ اﻟﻌﺎم ﯾﻜﻮن ‪٢ = α‬ﻥﺐ ± ‪β‬‬
‫اﻟﺒﺮھﺎن‬
‫اﻣﺎ‬
‫ﺉ ‪٢ = α‬ﻥﺐ + ‪β‬‬
‫ﺟﺘﺎ ‪ = α‬ﺟﺘﺎ‪β‬‬
‫ﺉ ‪ = α‬ﻋﺪد زوﺟﻰ×ﺐ + ‪β‬‬
‫او‬
‫ﺟﺘﺎ ‪ = α‬ﺟﺘﺎ)٢ﺐ - ‪ (β‬ﺉ ‪٢ = α‬ﻥﺐ + ٢ﺐ - ‪β‬‬
‫ﺉ ‪٢) = α‬ﻥ+٢(ﺐ - ‪β‬‬
‫ﺉ ‪ = α‬ﻋﺪد زوﺟﻰ×ﺐ - ‪β‬‬
‫ﺇ اﻟﺤﻞ اﻟﻌﺎم ﯾﻜﻮن ‪٢ = α‬ﻥﺐ ± ‪β‬‬
‫ﻣﺜﺎل ﺗﻄﺒﯿﻘﻰ‬
‫أوﺟﺪ اﻟﺤﻞ اﻟﻌﺎم ﻟﻠﻤﻌﺎدﻟﺔ: ﺟﺎ٢‪ = θ‬ﺟﺘﺎ ‪θ‬‬
‫اﻟﺤﻞ‬
‫ﺐ‬
‫٢‬
‫ﺟﺘﺎ) - ٢‪ = (θ‬ﺟﺘﺎ ‪θ‬‬
‫ﺐ‬
‫‪٢ = θ‬ﻥﺐ ± ) - ٢‪(θ‬‬
‫اﻣﺎ ‪٢ = θ‬ﻥﺐ‬
‫٢‬
‫ﺐ‬
‫+)‬
‫ﺐ٢‬
‫٣‪٢ = θ‬ﻥﺐ +‬
‫او‬
‫- ٢‪(θ‬‬
‫٢ﻥﺐ‬
‫+‬
‫ﺉ ‪=θ‬‬
‫٦‬
‫٣‬
‫٢‬
‫ﺐ‬
‫‪٢ = θ‬ﻥﺐ - ) - ٢‪(θ‬‬
‫٢‬
‫ﺐ‬
‫‪٢ = θ‬ﻥﺐ - ٢ + ٢‪θ‬‬
‫ﺐ‬
‫ﺉ‬
‫‪٢= θ‬ﻥﺐ -‬‫٢‬
‫اﻋﺪاد اﻷﺳﺘﺎذ / اﺑﺮاھﯿﻢ اﻷﺣﻤﺪى اﺑﺮاھﯿﻢ‬
‫ﻣﻌﻠﻢ ﺧﺒﯿﺮ اﻟﺮﯾﺎﺿﯿﺎت‬
‫ﺐ‬
‫‪٢- = θ‬ﻥﺐ +‬
‫ﺐ‬
‫٢‬
‫2110 390 2110‬

3. ‫3‬
‫‪www.mathtop4.alafdal.net‬‬
‫ﺛﺎﻟﺜﺎ:اﻟﻤﻌﺎدﻟﺔ اﻟﺘﻰ ﻋﻠﻰ اﻟﺼﻮرة ظﺎ ‪ = α‬ظﺎ‪β‬‬
‫ﻣﻨﺘﺪى اﻟﺮﯾﺎﺿﯿﺎت اﻟﻤﺘﻘﺪﻣﺔ‬
‫اﻟﺤﻞ اﻟﻌﺎم ﯾﻜﻮن ‪ = α‬ﻥﺐ + ‪β‬‬
‫اﻟﺒﺮھﺎن‬
‫اﻣﺎ‬
‫ﺉ ‪٢ = α‬ﻥﺐ + ‪β‬‬
‫ظﺎ ‪ = α‬ظﺎ‪β‬‬
‫ﺉ ‪ = α‬ﻋﺪد زوﺟﻰ×ﺐ + ‪β‬‬
‫او ظﺎ ‪ = α‬ظﺎ)ﺐ + ‪ (β‬ﺉ ‪٢ = α‬ﻥﺐ + ﺐ + ‪β‬‬
‫ﺉ ‪٢) = α‬ﻥ+1(ﺐ + ‪β‬‬
‫ﺉ ‪ = α‬ﻋﺪدﻓﺮدى+ ‪β‬‬
‫ﺇ اﻟﺤﻞ اﻟﻌﺎم ﯾﻜﻮن ‪ = α‬ﻥﺐ + ‪β‬‬
‫ﻣﺜﺎل ﺗﻄﺒﯿﻘﻰ‬
‫أوﺟﺪ اﻟﺤﻞ اﻟﻌﺎم ﻟﻠﻤﻌﺎدﻟﺔ: ظﺎ٢‪ = θ‬ظﺘﺎ‪θ‬‬
‫اﻟﺤﻞ‬
‫ظﺎ٢‪ = θ‬ظﺎ )‬
‫٢‪ = θ‬ﻥﺐ + )‬
‫٣‪= θ‬‬
‫ﺐ‬
‫٢‬
‫ﺐ‬
‫‪(θ‬‬‫-‪(θ‬‬
‫٢ ﺐ‬
‫ﻥﺐ +‬
‫٢‬
‫ﻥﺐ‬
‫+‬
‫‪=θ‬‬
‫٦‬
‫٣‬
‫ﺐ‬
‫اﻋﺪاد اﻷﺳﺘﺎذ / اﺑﺮاھﯿﻢ اﻷﺣﻤﺪى اﺑﺮاھﯿﻢ‬
‫ﻣﻌﻠﻢ ﺧﺒﯿﺮ اﻟﺮﯾﺎﺿﯿﺎت‬
‫2110 390 2110‬