Формули в геометрията Formulas in geometry Тригонометрични функции на обобщен ъгъл Trigonometric functions of the generalized angle

Градyси 0° 30° 45° 60° 90° 120° 135° 150° 180° 270° 360° Радиани Radians π 6 4 3 2π 3π 5π 2 Примери(Examples): 1) 2) sin 7π = sin 7.180°= cos 9π = cos 9°.180° = 3 3 4 4 = sin 420° = (60°+360°) = = cos 405° =(45°+360°) = = sin 60° = √3 = cos 45°= √ 2 2 2

3) 4) sin(-x).cos(-x).cotg(-x) = cos(- π ) = cos(+45°) = cos(2π-x) 4 -sinx.cosx.(-cotgx) = = √2 cosx 2 sinx.cosx = cosx sinx tg(2π) = tg120° = √3 3

α 0 ° 30° 45° 60° 90° 120° 135° 150° 180° Sin 1 2 √2 √3 Cos - √2 - √3 -1 Tg 3 -√3 Cotg

ОСНОВНИ ЗАДАЧИ и функции Main tasks and functions Sin(150°)= sin150° = 1 tg(60°) = tg60° = √3 2 Cos(135°) = cos135° = -√2 cotg(90°) = cotg90° = 0 sinα = a cosα = b tgα = a cotgα = b c c b a sinβ = b cosβ = a tgβ= b cotgβ= a c c a b

Sin(90°-α) = cosα Cos(90°-α) = sinα Tg(90°-α) = cotgα Cotg(90°-α) = tgα Sin(90°+α)=cosα Cos(90°+α)=-sinα Tg(90°+α)=-cotgα Cotg(90°+α)=-tgα Sin(180°-α)=sinα Cos(180°-α) = - cosα Tg(180°-α)=-tgα Cotg(180°-α)=-cotgα Sin(180°+α)=-sinα Cos(180°+α)=-cosα Tg(180°+α)=tgα Cotg(180°+α)=cotgα Sin(270°-α)=-cosα Cos(270°-α)=-sinα Tg(270°-α)=cotgα Cotg(270°-α)=tgα Sin(270°+α)=-cosα Cos(270°+α)=sinα Tg(270°+α)=-cotgα Cotg(270°+α)=-tgα

sin300° = sin(30 °+270 °) = sin30 ° = 1 2 cos240° = cos(60°+180°) = cos60°= 1 Tg(-1410° ) = -tg(3x360°+330°) = -tg330° = = -tg(-30°+360°) = tg30° = √3 3 Cotg(750°) = cotg(30°+2.360°) = cotg30° = = √3

Sin(-α) = -sinα tg(-α) = -tgα Cos(-α)= cosα cotg(-α) = -cotgα sin²α+cos²α = 1 tgα=sinα cosα cotgα = cosα sinα tgα.cotgα = 1 Sin(α+k.360°) = sinα Cos(α+k.360°) = cosα Tg(α+k.360°) = tgα k=0,±1,±2,… Cotg (α+k.360°) = cotgα Sin(-α) = -sinα tg(-α) = -tgα Cos(-α)= cosα cotg(-α) = -cotgα Tg(α+k.180°) = tgα cotg(α + k.180°) = cotgα k=0,±1,±2,…

Пресметнете Estimate Cos110°.cos50°+ sin110°.sin50°= cos(110°-50°) = cos60° = 1 2 sin50°.cos20° - sin20°.cos50° = sin(50°-20°) = sin30° = 1 cos20°.cos70° - sin20°.sin70° = cos(20°+70°) = cos90° = 0 tg63°-tg33° = tg30° = √3 1 + tg63°.tg33° 3 sin65°.cos25° + sin25°.cos65° = sin(65°+25°) = sin90° = 1

Опростете изразите Simplify expressions cos(30°+α) – cos(30°-α) = cos30°.cosα-sin30°.sinα - [ cos30°.cosα + sin30°sinα] = cos30°cosα – sin30°sinα – cos30°cosα – sin30°sinα = = -2sin30°sinα = -2 . 1 sinα = - sinα 2 Sin(α-30°) + cos( 60° - α) = sinα.sin30° - cosα.cos30° + cos60°.cosα + sin60°.sinα = = √3 cos - √3 cosα + 1 2 3 2 tg(π+x)cos(-x)cotg 3π = tgx.cosx(-cotg270°) = sinx . cosx . cotg(90°+180°) = 2 sin90° cosx sin π = sinx.cotg90° = (sinx).0 = 0

Други примери Other examples ∆ABC, <C=90 °, c=10cm , tgα = 3 B 4 tg=a = 3 a= 3 . b a c=10cm b 4 4 C b A a² + b² = c² - Питагорова теорема Р-е: a² + b² = c² b² = 100. 16 b = √64 (3 b) ² + b² = 10² 25 4 b = 8cm 9 b² + b² = 100 a = 3 . b = 3 . 8 = 6cm 16 4 4 1 9 .b² = 100 sinα = a = 6 = 0,6cm 16 c 10 25 . b²= 100 cosα = b = 8 = 0,8cm 16 c 10

∆ABC , AC=BC=10cm , sinα =4 , S =? C 5 Р-е : (Pythagorean theorem) 10 10 ∆AHC – правоъг. = Питагорова теорема а²+b²=c² AH² + CH² = AC² h²+a² =c² a² + 8² = 10² A a H a B a² = 100 – 64 = 36 a = 6cm sinα = h AC AB = 2.6 = 12cm h = 4 10 5 S = AB.CH = 12.8 = 48cm² h = 4 . 10 = 8cm 2 2 5

Тригонометрични функции на двоен (удвоен) ъгъл Trigonometric functions of double (geminate) angle Sin3α=sinα(3-4sin²α) Cos3α=cosα(4cos²α-3) Sin2α = 2sinα .cosα Cos2α = cos²α-sin²α 2cos²α – 1 1-2sin²α Tg2α = 2tgα 1 - tg²α Cotg2α=cotg²α – 1 2cotgα sin α = ± √ 1 - cosα 2 2 cos α = ± √ 1 +cosα tg α = ± √ 1 – cosα 2 1 + cosα cotg α = ± √ 1 + cosα 2 1 - cosα

Приложение на формулите Application of formulas sin36 ° = sin2.18 ° = 2sin18 °cos18 ° cos18 °cos36 °= 2.sin18 °cos18 °cos36 °=sin2.18 °cos36 ° = 2sin18 ° 2sin18 ° 2sin36 °cos36 ° = sin72 ° = sin(90 °-18 °) = cos18 ° = 1 cotg18 ° 2.2sin18 ° 4sin18 ° 4sin18 ° 4sin18° 4 cos36 °cos72 °=2sin36 °cos36 °cos72 ° = 2sin72 °cos72 ° = 2sin36 ° 2.2sin36 ° = sin144 ° = sin(180 °-36 °) = sin36 ° = 1 4sin36 ° 4sin36 ° 4sin36 ° 4 2sin15 °cos15 ° = sin30 ° = 1 = 1 2 2 2.2 4

cos25°cos65°=cos(90°-65°) cos65°= 2sin65°.cos65 ° sin130° = 1 sin130° 2 2 2sin18°sin72° = 2sin18°sin(90°-18°) = 2sin18°cos18° =sin36° 2cos²12-1=cos24 ° cos²15-sin²15°=cos30 ° = √3 2cos²(45°– α) -1 = cos2(45°- α) = cos(90°-α) = sinα 2 2

Още задачи (More tasks): cosα = -3 ,α ε (π ; π); sinα ; sinα ; cosα ; tgα ; cotgα = ? 2 5 2 2 2 sin²α+cos²α = 1 2 2 sin α = ±√1-cos² α = √1 – (3)² = √ 1 – 9 = √25-9 = √ 16 = 4 2 2 5 25 25 25 5 Sin2α = 2sinα cosα Sinα=2sinα cosα = 2 . 4 (-3) sinα = -24 2 2 5 5 25 Cos2α=cos²α-sin²α Cosα = cos²α-sin²α = (-3)² - (4)² = 9 – 16 = -7 2 2 5 5 25 25 25 tgα = sinα = -24 : (- 7) = + 24 . 25 = tgα = 24 cotgα =7 cosα 25 25 25 7 7 24

sinα = -40 α ε (270° ; 360°) ~> IV квадрант(quadrant) 41 Cos = ? Sinα = ? ; sinα = ? 2

Представете като произведение (Presented as production ) 1+sinα = sin 90°+ sinα = 2sin 90°+α cos 90°- α = 2 2 = 2sin(45°+ α) cos (45° - α) 2 2 1 – sinα = sin90° - sinα = = 2sin 90°- α cos 90° + α = 2sin(45°– α) cos(45°+ α) 2 2 2 2 cos3α + cosα = = 2cos 3α+α cos 3α -2 = 2cos2α cosα = 2 2 = cos2α – cosα = -2sin2α+α sin 2α –α = -2sin3α sinα 2 2 2 2

Основни тъждества (Primary Identity) sinα + sinβ = 2sin α + β cos α – β , 2 2 sinα - sinβ = 2sin α - β cos α + β, 2 2 cosα + cosβ = 2cos α + β cos α - β cosα - cosβ = -2sin α + β sin α - β . 2 2 Sinα.cosβ = 1 [sin(α-β) +sin(α+β)], 2 Cosα.cosβ = 1 [cos(α-β) + cos(α+β)], Sinα.sinβ = 1 [cos(α-β) – cos(α+β)].

sin75°cos75° = 2sin75°cos75°=sin(2.75°) = sin150° = 1 = 1 2 2 2.2 4 sin15°cos15°= 2sin15°cos15° = sin(2.15°) = sin 30° = 1 = 1 2 2 2.2 4 sin165°= sin120.°cos45°+cos120°.sin45° = = √3 .√2 + (-1) . √2 = √3.√2 - √2 = √2(1-√3) 2 2 2 2 4 4 4 cos105° = cos45°.cos60 °– sin45°.sin60° = = √2 . 1 - √2 . √3 = √2 - √2.√3 = √2(1-√3) 2 2 2 2 4 4 4 sin105°.cos15° - cos105°.sin15° = = sin(105°-15°) = sin90° = 1

√2 – 2sinα = 2(√2 – sinα) = 2(sin45° - sinα) = = 2(sin45°-sinα) = 2(2sin 45°+α cos 45° - α) = 2 2 = 4 sin45°+α cos 45°- α 2 2 2sinα +√3 = 2(sinα +√3) = 2(√3 + sinα) = 2 2 = 2(sin60° + sinα) = 2(2sin 60°+α cos 60°-α) = = 2(2sin( 30°+α ) cos( 30° - α) = 2 2 = 4sin (30°+ α) cos (30°- α) 2 2

cosα + cos5α - cos2α - cos4α = 2cos 6α cos 4α – (cos2α + cos4α) = 2 2 = 2cos 6α cos 4α – 2cos 6α cos(-2α) = 2 2 2 2 = 2cos 6α (4α – cos2α) = 2cos 3α (cos2α - cosα) = 2 2 2 = 2cos3α(-2 sin 3α sin α) = 4cos 3α sin 1,5α sin 0,5α sin12°cos48° + cos12°sin48° = sin (12°+48°) = sin60° = √3 2 cos78° cos 18° + sin78° sin 18° = cos (78° - 18°) = cos60° = 1

1+cos2α + sin2α = cos(30°+α) - cos(30°-α) = cos30°.cosα – sin30°.sinα – [cos30°cosα+sin30°sinα] = = cos30°cosα – sin30°sinα – cos30°cosα – sin30°sinα = -2sin30°sinα = = -2.1sinα = -sinα 2 cos(45°+α) – cos(45°-α) = cos45°.cosα – sin45°.sinα – [cos45°cosα+sin45°sinα] = = cos45°cosα – sin45°sinα – cos45°cosα – sin45°sinα = -2sin45°sinα = = -2.√2sinα = -√2sinα Sin(60°+α) – sin(60°-α) = sin60°cosα+ cos60°sinα – [sin60°cosα – cos60°sinα] = = sin60°cosα + cos60°sinα – sin60°cosα + cos60°sinα = 2cos60°sinα = = 2. 1sinα = sinα

По инициатива на г-жа Маргарита Малинова (At the initiative of Mrs По инициатива на г-жа Маргарита Малинова (At the initiative of Mrs. Margarita Malinova) Made by: Vladislav Denev Tihomoir Jelev Magdalena Veleva Desislava Petrova