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Тригонометрични функции на обобщен ъгъл Trigonometric functions of the generalized angle.

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Presentation on theme: "Тригонометрични функции на обобщен ъгъл Trigonometric functions of the generalized angle."— Presentation transcript:

1 Тригонометрични функции на обобщен ъгъл Trigonometric functions of the generalized angle

2 Град y си 0°0°30°45°60°90°120°135°150°180°270°360° Радиани Radians 0 π 6 π4π4 π3π3 π4π4 2π32π3 3π43π4 5π65π6 π 3π23π2 2π2π

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4 α 0 °30°45°60°90°120°135°150°180° Sin √2 2 √3 2 1 √3 2 √ Cos 1√3 2 √ √2 2 - √3 2 Tg0√3 3 1√3√3-√3-√3-√3 3 0 Cotg√3√31√3 3 0-√3 3 -√3-√3

5 Sin(150°)= sin150° = 1 tg(60°) = tg60° = √3 2 Cos(135°) = cos135° = -√2 cotg(90°) = cotg90° = 0 2 sin α = a cos α = b tg α = a cotg α = b c c b a sin β = b cos β = a tg β = b cotg β = a c c a b

6 Sin(90°- α ) = cos α Cos(90°- α ) = sin α Tg(90°- α ) = cotg α Cotg(90°- α ) = tg α Sin(90°+ α )=cos α Cos(90°+ α )=-sin α Tg(90°+ α )=-cotg α Cotg(90°+ α )=-tg α Sin(180°- α )=sin α Cos(180°- α ) = - cos α Tg(180°- α )=-tg α Cotg(180°- α )=-cotg α Sin(180°+ α )=-sin α Cos(180°+ α )=-cos α Tg(180°+ α )=tg α Cotg(180°+ α )=cotg α Sin(270°- α )=-cos α Cos(270°- α )=-sin α Tg(270°- α )=cotg α Cotg(270°- α )=tg α Sin(270°+ α )=-cos α Cos(270°+ α )=sin α Tg(270°+ α )=-cotg α Cotg(270°+ α )=-tg α

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

8 Sin( α +k.360°) = sin α Cos( α +k.360°) = cos α Tg( α +k.360°) = tg α k=0,±1,±2,… Cotg ( α +k.360°) = cotg α Sin(- α ) = -sin α tg(- α ) = -tg α Cos(- α )= cos α cotg(- α ) = -cotg α Tg( α +k.180°) = tg α cotg( α + k.180°) = cotg α k=0,±1,±2,… sin² α +cos² α = 1tg α =sin α cos α cotg α = cos α sin α tg α.cotg α = 1

9 Cos110°.cos50°+ sin110°.sin50°= cos(110°-50°) = cos60° = 1 2 sin50°.cos20° - sin20°.cos50° = sin(50°-20°) = sin30° = 1 2 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

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

11 ∆ ABC,

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

13 Sin3 α =sin α (3-4sin²α) Cos3 α =cos α (4cos²α-3) Sin2 α = 2sin α.cos α Cos2 α = cos² α -sin² α 2cos² α – 1 1-2sin² α Tg2 α = 2tg α 1 - tg² α Cotg2 α =cotg² α – 1 2cotg α sin α = ± √ 1 - cosα 2 2 cos α = ± √ 1 +cosα 2 2 tg α = ± √ 1 – cosα cosα cotg α = ± √ 1 + cosα cosα

14 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 =

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

16 cos α = -3, α ε ( π ; π); sin α ; sin α ; cos α ; tg α ; cotg α = ? sin²α+cos²α = sin α = ±√1-cos² α = √1 – (3)² = √ 1 – 9 = √25-9 = √ 16 = Sin2 α = 2sin α cos α Sin α =2sin α cos α = 2. 4 (-3) sin α = Cos2 α =cos² α -sin² α Cosα = cos²α-sin²α = (-3)² - (4)² = 9 – 16 = tg α = sin α = -24 : (- 7) = = tg α = 24 cotg α =7 cos α

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

18 1+sin α = sin 90°+ sin α = 2sin 90°+ α cos 90°- α = 2 2 = 2sin(45°+ α ) cos (45° - α ) – sin α = sin90° - sin α = = 2sin 90°- α cos 90° + α = 2sin(45°– α ) cos(45°+ α ) cos3α + cos α = = 2cos 3 α + α cos 3 α -2 = 2cos2 α cos α = 2 2 = cos2 α – cos α = -2sin2 α +α sin 2 α – α = -2sin3 α sin α

19 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 2 2 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)], 2 Sinα.sinβ = 1 [cos(α-β) – cos(α+β)]. 2

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

21 √2 – 2sin α = 2(√2 – sin α ) = 2(sin45° - sin α ) = 2 = 2(sin45°-sin α ) = 2(2sin 45°+ α cos 45° - α ) = 2 2 = 4 sin45°+ α cos 45°- α 2 2 2sin α +√3 = 2(sin α +√3) = 2(√3 + sin α ) = 2 2 = 2(sin60° + sin α ) = 2(2sin 60°+ α cos 60°-α) = 2 2 = 2(2sin( 30°+α ) cos( 30° - α ) = 2 2 = 4sin (30°+ α) cos (30°- α) 2 2

22 cos α + cos5 α - cos2 α - cos4 α = 2cos 6 α cos 4 α – (cos2 α + cos4 α ) = 2 2 = 2cos 6 α cos 4 α – 2cos 6 α cos(-2 α ) = = 2cos 6 α (4 α – cos2 α ) = 2cos 3 α (cos2 α - cos α ) = = 2cos3 α (-2 sin 3 α sin α ) = 4cos 3 α sin 1,5 α sin 0,5 α 2 2 sin12°cos48° + cos12°sin48° = sin (12°+48°) = sin60° = √3 2 cos78° cos 18° + sin78° sin 18° = cos (78° - 18°) = cos60° = 1 2

23 1+cos2 α + sin2 α = cos(30°+ α ) - cos(30°- α ) = cos30°.cos α – sin30°.sin α – [cos30°cos α +sin30°sin α ] = = cos30°cos α – sin30°sin α – cos30°cos α – sin30°sin α = -2sin30°sin α = = -2.1sin α = -sin α 2 cos(45°+ α ) – cos(45°- α ) = cos45°.cos α – sin45°.sin α – [cos45°cos α +sin45°sin α ] = = cos45°cos α – sin45°sin α – cos45°cos α – sin45°sin α = -2sin45°sin α = = -2.√2sin α = -√2sinα 2 Sin(60°+ α ) – sin(60°- α ) = sin60°cos α + cos60°sinα – [sin60°cos α – cos60°sinα] = = sin60°cos α + cos60°sin α – sin60°cos α + cos60°sinα = 2cos60°sin α = = 2. 1sin α = sinα 2

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