أبنائي الطلبة:

لقد رأيت أن أضع هنا عدة مسائل في الفصلين الأخيرين من مادة الـ trig تاركاً حلها للطلبة وذلك لفترة مؤقتة، وعلي الطالب أن يسألني عن بعض المسائل التي استعصي عليه حلها... ثم بعد فترة سوف أقوم بحل تلك الأسئلة لتكون مرجعاً للطالب ليلة الإمتحان، وهذه المسائل بعضها موجود في كتاب الـ evaluation والبعض الآخر في كتاب الـ work book أما البعض الثالث فهو موجود في إمتحانات آخر العام وذلك ليختبر الطالب نفسه ومدي تحصيله للمادة...

وإن شاء الله تعالي سوف أبدأ مستعيناً بالله فهو حسبنا ونعم الوكيل

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 1 \right )\, without\, \, using\, \, calculator\, \, prove\, \, that:\, \, sin54-sin18=\frac{1}{2}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 2 \right )\, \, prove\, \, that:\, sin\left ( a+b \right )\cdot sin\left ( a-b \right )=sin^2a-sin^2b


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 3 \right )\, if\, \, sin\left ( a+b \right )=\frac{3}{4},\, \, sin\left ( a-b \right )=\frac{1}{2}\, \, prove\, \, that:\, tan\left ( a \right )=5tan\left ( b \right )


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 4 \right )\, \, prove\, \, that:\, \, tan^215+2\sqrt{3}tan15-1=0,\, \, and\, \, hence\, prove\, \, that\, \, tan15=2-\sqrt{3}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 5 \right )\, \, if\, \, tan\left ( c \right )+sec\left ( c \right )=\frac{3}{2}\, such\, \, that\, \, \left ( c \right )\, \, is\, \, an\, \, acute\, \, angle,
http://latex.codecogs.com/gif.latex?\dpi{120} find\, \, the\, \, value\, \, of:\, \, 1+cos\left ( 2c \right )


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 6 \right )\, \, without\, \, using\, \, calculator\, \, prove\, \, that:\, \, \frac{sin65+sin25}{cos40+cos80}=\sqrt{2}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 7 \right )\, \, prove\, \, that:\, cos^4a=\frac{1}{8}cos\left ( 4a \right )+\frac{1}{2}cos\left ( 2a \right )+\frac{3}{8}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 8 \right )\, \, prove\, \, that:\, \, \frac{cos\left ( a \right )+sin\left ( a \right )}{cos\left ( a \right)-sin\left ( a \right )}=\frac{1}{cos\left ( 2a \right )}+tan\left ( 2a \right )


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 9 \right )\, \, prove\, \, that:\, \, cosec18-cosec54=2


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 10 \right )\, in\, \, the\, \, triangle\, \, abc\, \, prove\, \, that:\, \, a^2=b^2+c^2-2b\cdot c\cdot cos\left ( A \right )

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 11 \right )\, \, if\, \, \frac{cos\left ( B \right )}{a}=\frac{cos\left ( A \right )}{b}\, \, in\, \, the\, \, triangle\, \, abc,\, \, prove\, \, that\, this\, \, triangle\, \, is:
http://latex.codecogs.com/gif.latex?\dpi{120} either\, \, right\, \, angled\, \, triangle\, \, or\, \, isosceles\, \, triangle


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 12 \right )\, \, prove\, \, that:\, \, sinA+sinB+sinC=\frac{4P\bigtriangleup }{a.b.c}
http://latex.codecogs.com/gif.latex?\dpi{120} such\, \, that:\, \, P\, \, is\, \, the\, \, half\, \, of\, \, its\, \, peremiter\, \, and\, \, \bigtriangleup is\, \, its\, \, surface\, \, area



http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 13 \right )\, \, in\, \, the\, \, previous\, \, problem\, \, prove\, \, that:
http://latex.codecogs.com/gif.latex?\dpi{120} \frac{1}{\left ( c \right )sinA}=\frac{1}{\left ( a \right )sinB}=\frac{1}{\left ( b \right )sinC}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 14 \right )\, without\, \, using\, \, calculator,\, prove\, \, that:
http://latex.codecogs.com/gif.latex?\dpi{120} \left ( sin5+cons5 \right )\left ( sin40+cos50 \right )=\sqrt{2}cos10


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 15 \right )\, \, without\, \, calculator\, \, prove\, \, that:
http://latex.codecogs.com/gif.latex?\dpi{120} tan75=1+tan30+tan30.tan75

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 16 \right )\, prove\, \, that:\, \frac{1-tanx}{1+tanx}=\frac{cos2x}{1+sin2x}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 17 \right )abc\, is\, \, a\, \, triangle\, \, which\, \, a=12cm,\, \, b=10cm,\, \, and\, \, c=8cm
http://latex.codecogs.com/gif.latex?\dpi{120} prove\, \, that:\, \,m\left ( A \right )=2m\left ( C \right )


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 18 \right )\, prove\, \, that:\, cos\frac{A}{2}=\pm \sqrt{\frac{1+cosA}{2}}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 19 \right )\, without\, \, using\, \, calculator\, \, prove\, \, that:\, \left ( cos20 \right )\left ( cos40 \right )\left ( cos80 \right )=\frac{1}{8}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 20 \right )\, prove\, \, that:\, \, 4\left ( sin^6x+cos^6x \right )=1+3cos^22x

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 21 \right )\, prove\, \, that:\, \, tan2x\cdot secx=2sinx\cdot sec2x


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 22 \right )\, find\, \, the\, \, min.\, \, and\, \, max.\, \, value\, \, of:\, sinx\cdot cos70+cosx\cdot sin70
http://latex.codecogs.com/gif.latex?\dpi{120} and\, \, find\, \, \left ( x \right )\, at\, \, every\, \, case:\, such\, \, that:\, \, 0^{\circ}\ll x< 360^{\circ}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 23 \right )\, find\, \, the\, \, max.\, \, and\, \, min.\, \, value\, \, of:\, \, 3sinx\cdot cos105-3sinx\cdot sin105
http://latex.codecogs.com/gif.latex?\dpi{120} and\, \, find\, \, \left ( x \right )\, at\, \, every\, \, case\, s.t\, \, 0^{\circ}\ll x< 360^{\circ}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 24 \right )\, in\, \bigtriangleup abc,\, \, prove\, \, that:

http://latex.codecogs.com/gif.latex?\dpi{120} sin\left ( \frac{a+b-c}{2} \right )\cdot cos\left ( a+b \right )-cos\left ( \frac{a+b-c}{2} \right )\cdot \cdot sin\left ( a+b \right )=-1


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 25 \right )\, if\, cosx-cos2x+cos3x=0,\, \, find\, \left ( x \right )\, s.t:\, 0^{\circ}\ll x< 360^{\circ}

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 26 \right )\, \, evaulate:\, \, sin75+sin15


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 27 \right )\, \, evalute:\, tan\frac{\pi }{12}\times tan\frac{7\pi }{12}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 28 \right )\, prove\, \, that:\, tan70-tan25-tan70\cdot tan25=1


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 29 \right )\, \, prove\, that:\, sin22+sin38=sin98


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 30 \right )\, \, prove\, that:\, cos\left ( 2A \right )-4cos\left ( A \right )+3=8sin^4\frac{A}{2}

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 31 \right )\, \, if\, 4sin\left ( 2x \right )+3cos\left ( 2x \right )=3,\, \, prove\, \, that:\, tan\left ( x \right )=\frac{4}{3}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 32 \right )\, in\, the\, \bigtriangleup abc,\, if\, a=2b,\, prove\, \, that:\, a^2=b\left ( c+b \right )


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 33 \right )\, if\, sin\left ( C \right )+cos\left ( C \right )=1.24,\, find\, \, each\, \, of:\, \, sin\left ( 2C \right )\, \, and\, \, tan\left ( C \right )+cot\left ( c \right )


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 34 \right )\, at\, any\, not\, \, right\, \, angled\, \, triangle\, abc,\, prove\, \, that:
http://latex.codecogs.com/gif.latex?\dpi{120} its\, \, area\, =\frac{1}{4}\left ( a^2+b^2-c^2 \right )\times tan\left ( C \right )


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 35 \right )\, in\, \bigtriangleup abc,\, if:\, \, log\left ( a^2 \right )=log\left ( b^2 \right )-log\left ( 2b\cdot c\cdot cos\left ( A \right )\right
http://latex.codecogs.com/gif.latex?\dpi{120} prove\, that\, \bigtriangleup abc\, \, is\, \, isosceles\, \, triangle

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 36 \right )\, in\, \, any\, \, \bigtriangleup \, \, prove\, \, that\, \, its\, \, aria=\frac{b^2\cdot sinA\cdot sinC}{2sinB}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 37 \right )\, \, in\, the\, \bigtriangleup abc,\, m\left ( BAC \right )=65^{\circ},\, \, \bar{bd}\, \, bisects\, \, \angle \left ( ABC \right )\, \, and\, \, cuts\, \, \bar{ac}\, at\, \, \left ( d \right )
http://latex.codecogs.com/gif.latex?\dpi{120} if\, \, ad=6cm,\, \, dc=10cm,\, \, find\, \, the\, \, length\, \, of\, \, \bar{ab}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 38 \right )\, \, if\, sinC\cdot cos^3C-sin^3C\cdot cosC=\frac{1}{4},\, find\, \, the\, \, value\, \, of\, \, sin4C


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 39 \right )in\, \, the\, \bigtriangleup \, abc,\, if\, \, a.b.cosC+b.c.cosA=a^2+c^2,\, prove\, \, that:\, m\left ( \hat{B} \right )=90^{\circ}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 40 \right )\, if\, \, tanA=\left ( \sqrt{2}+1 \right )\cdot tanx,\, tanB=\left ( \sqrt{2}-1 \right )\cdot tanx
http://latex.codecogs.com/gif.latex?\dpi{120} prove\, \, that:\, tan\left ( a-b \right )=sin2x

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 41 \right )\, in\, \bigtriangleup abc,\, if\, c=\sqrt{a^2+a.b+b^2},\,
http://latex.codecogs.com/gif.latex?\dpi{120} find\,\, the\,\, measure\, of\, \, the\, \, greatest\, \, angle\, \, in\, \bigtriangleup


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 42 \right )\, solve\, \, the\, \, trig.\, \, equation:\, sin2x=cos3x:\, 0< x< \pi


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 43 \right )\, solve\, \, the\, \, equation:\,\, sinx+\sqrt{3} cosx=1,\, 0< x< 2\pi


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 44 \right )\, prove\, \, that:\, \frac{sin^23x}{sin^2x}-\frac{cos^23x}{cos^2x}=8cos2x


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 45 \right )\, if:\, 3sinA+4cosB=6,\, \, and:\, \, 4sinB+3cosA=1,\,
http://latex.codecogs.com/gif.latex?\dpi{120} find\, \, m\left ( C^{\circ} \right )\, \, knowing\, \, that\, \, \bigtriangleup abc\, \, is\, \, acute\, \, angled\, \, triangle

http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 46 \right )\, in\, \bigtriangleup abc:\, prove\, \, that:\, \frac{sin\left ( A-B \right )}{sin\left ( A+B \right )}=\frac{a^2-b^2}{c^2}


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 47 \right )\, if\, \, the\, \, area\, \, of\, \, a\, \, triangle\, \, is\, \, 10\sqrt{3}\, cm^2\, and\, \, its\, \, perimeter\, \, is\, \, 20cm
http://latex.codecogs.com/gif.latex?\dpi{120} and\, \, the\, \, measure\, \, of\, \, an\, \, angle\, \, in\, \, it\, \, is\, \, 60^{\circ},\, \, find\, \, the\, \, length\, \, of\, \, its\, \, sides


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 48 \right )\, in\, \bigtriangleup abc:\, \, if\, \, m\left ( B^{\circ} \right )=m\left ( C^{\circ} \right ),\, and\, \, if:\, sin\frac{A}{2}\cdot sin\frac{B}{2}\cdot sin\frac{C}{2}=\frac{1}{8}
http://latex.codecogs.com/gif.latex?\dpi{120} prove\, \, that\, \, \bigtriangleup abc\, \, is\, \, equilateral


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 49 \right )\, prove\, \, that:\, tan\left ( 3\theta \right )=\frac{3tan\left ( \theta \right )-tan^3\left ( \theta \right )}{1-3tan^2\left ( \theta \right )},\, \, and\, \, hence:\, \, prove\, \, that:
http://latex.codecogs.com/gif.latex?\dpi{120} tan20\cdot tan30\cdot tan40=tan10


http://latex.codecogs.com/gif.latex?\dpi{120} \left ( 50 \right )\, \, prove\, \, that:\, \, tan50\cdot tan60\cdot tan70=tan80

وهكذا أبنائي الطلاب فقد انتهينا من تغطية الفصلين الأخيرين من الـ trig وأنا منتظر منكم حلول تلك المسائل، وكما أكدت سابقاً، فعلي كل من يتعثر في أي مسألة من المسائل السابقة منكم يا أبنائي أن يخبرني علي صفحات هذا المنتدي لكي أقوم بحلها أو يقوم أحد الأساتذة الكرام بحلها له، فكلنا نكمل بعضنا البعض كمعلمين ونريد في النهاية الخير والتفوق لأبنائنا الطلاب.

منتظرين تعليقاتكم وتساؤلاتكم

مجهود كبير من أستاذ عظيم معطاء كريم

بارك الله فيك يا أستاذى وزادك من علمه

http://alyeldeen.com/vb/progs4u/misc/quotes/quot-top-left.gifاقتباس:http://alyeldeen.com/vb/progs4u/misc/quotes/quot-top-right.gifhttp://alyeldeen.com/vb/progs4u/misc/quotes/quot-by-left.gifالمشاركة الأصلية كتبت بواسطة الاستاذ على الدين يحيىhttp://alyeldeen.com/vb/progs4u/misc/quotes/quot-by-right.gifhttp://alyeldeen.com/vb/progs4u/misc/quotes/quot-top-right-10.gif مجهود كبير من أستاذ عظيم معطاء كريم

بارك الله فيك يا أستاذى وزادك من علمه

http://alyeldeen.com/vb/progs4u/misc/quotes/quot-bot-left.gifhttp://alyeldeen.com/vb/progs4u/misc/quotes/quot-bot-right.gifوبارك فيك أستاذي العظيم وزادك بسطة في العلم والرزق ونفعك ونفع الأجيال القادمة بك كما نفع السابقة منها بغزارة علمكم وكرم خلقكم.

thanksssssssssssssssssssssssssssssssssssss
it is very good work

thanksssssssssssssssssssssssssssssssssssss
it is very good work

http://up.arabsgate.com/u/7221/5585/87123.gif

http://up.arabsgate.com/u/7221/5585/87137.gif

http://up.arabsgate.com/u/7221/5585/87152.gif

http://up.arabsgate.com/u/7221/5585/87162.gif

بارك الله فيكم

good work

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(http://scrapermat.ru)Kari (http://screwingunit.ru)Stan (http://seawaterpump.ru)Jame (http://secondaryblock.ru)Ever (http://secularclergy.ru)Kath (http://seismicefficiency.ru)Ajax (http://selectivediffuser.ru)Simo (http://semiasphalticflux.ru)Rick (http://semifinishmachining.ru)CARD (http://spicetrade.ru)CARD (http://spysale.ru)CARD (http://stungun.ru)Moir (http://tacticaldiameter.ru)Eber (http://tailstockcenter.ru) Atha (http://tamecurve.ru)Fiel (http://tapecorrection.ru)foam (http://tappingchuck.ru)Goud (http://taskreasoning.ru)Comp (http://technicalgrade.ru)Char (http://telangiectaticlipoma.ru)Cumm (http://telescopicdamper.ru)Aero (http://temperateclimate.ru)Erin (http://temperedmeasure.ru)Roal (http://tenementbuilding.ru)tuchkas (http://tuchkas.ru/)Lefe (http://ultramaficrock.ru)Rotr (http://ultraviolettesting.ru)