Тригонометрични функции на обобщен ъгъл

Trigonometric functions of the generalized angle

Градyси 0° 30° 45° 60°

90° 120°

135°

150°

180°

270°

360°

РадианиRadians

0 π 6

π4

π3

π4

2π3

3π4

5π6

π 3π2

2π

α 0 ° 30° 45° 60° 90° 120° 135° 150° 180°

Sin 0 1 2

√2 2

√3 2

1 √3 2

√2 2

1 2

0

Cos 1 √3 2

√2 2

1 2

0 -1 2

- √2 2

- √3 2

-1

Tg 0 √3 3

1 √3 -√3 -1 -√3 3

0

Cotg √3 1 √3 3

0 -√3 3

-1 -√3

Sin(150°)= sin150° = 1 tg(60°) = tg60° = √3 2Cos(135°) = cos135° = -√2 cotg(90°) = cotg90° = 0 2 sinα = a cosα = b tgα = a cotgα = b c c b asinβ = b cosβ = a tgβ= b cotgβ= a c c a b

Sin(90°-α) = cosαCos(90°-α) = sinαTg(90°-α) = cotgαCotg(90°-α) = tgα

Sin(90°+α)=cosαCos(90°+α)=-sinαTg(90°+α)=-cotgαCotg(90°+α)=-tgα

Sin(180°-α)=sinαCos(180°-α) = - cosαTg(180°-α)=-tgαCotg(180°-α)=-cotgα

Sin(180°+α)=-sinαCos(180°+α)=-cosαTg(180°+α)=tgαCotg(180°+α)=cotgα

Sin(270°-α)=-cosαCos(270°-α)=-sinαTg(270°-α)=cotgαCotg(270°-α)=tgα

Sin(270°+α)=-cosαCos(270°+α)=sinαTg(270°+α)=-cotgαCotg(270°+α)=-tgα

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

Sin(α+k.360°) = sinαCos(α+k.360°) = cosαTg(α+k.360°) = tgα k=0,±1,±2,…Cotg (α+k.360°) = cotgα

Sin(-α) = -sinα tg(-α) = -tgαCos(-α)= cosα cotg(-α) = -cotgα

Tg(α+k.180°) = tgα cotg(α + k.180°) = cotgα k=0,±1,±2,…

sin²α+cos²α = 1

tgα=sinα cosα

cotgα = cosα sinα

tgα.cotgα = 1

Cos110°.cos50°+ sin110°.sin50°= cos(110°-50°) = cos60° = 1 2 sin50°.cos20° - sin20°.cos50° = sin(50°-20°) = sin30° = 1 2cos20°.cos70° - sin20°.sin70° = cos(20°+70°) = cos90° = 0

tg63°-tg33° = tg30° = √31 + tg63°.tg33° 3

sin65°.cos25° + sin25°.cos65° = sin(65°+25°) = sin90° = 1

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 2Sin(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 2= sinx.cotg90° = (sinx).0 = 0

∆ABC, <C=90 °, c=10cm , tgα = 3 B 4 tg=a = 3 a= 3 . ba 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 = 6cm16 4 41 9 .b² = 100 sinα = a = 6 = 0,6cm 16 c 1025 . b²= 100 cosα = b = 8 = 0,8cm16 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 = 10²A a H a B a² = 100 – 64 = 36 a = 6cmsinα = h AC AB = 2.6 = 12cmh = 410 5 S = AB.CH = 12.8 = 48cm²h = 4 . 10 = 8cm 2 2 5

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 2 2

tg α = ± √ 1 – cosα 2 1 + cosα

cotg α = ± √ 1 + cosα 2 1 - cosα

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 ° 2sin130° = 1 sin130° 2 2

2sin18°sin72° = 2sin18°sin(90°-18°) = 2sin18°cos18° =sin36°

2cos²12-1=cos24 °cos²15-sin²15°=cos30 ° = √3 22cos²(45°– α) -1 = cos2(45°- α) = cos(90°-α) = sinα 2 2

cosα = -3 ,α ε (π ; π); sinα ; sinα ; cosα ; tgα ; cotgα = ? 2 5 2 2 2 sin²α+cos²α = 1 2 2sin α = ±√1-cos² α = √1 – (3)² = √ 1 – 9 = √25-9 = √ 16 = 4 2 2 5 25 25 25 5Sin2α = 2sinα cosαSinα=2sinα cosα = 2 . 4 (-3) sinα = -24 2 2 5 5 25Cos2α=cos²α-sin²αCosα = cos²α-sin²α = (-3)² - (4)² = 9 – 16 = -7 2 2 5 5 25 25 25tgα = 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)

41Cos = ? Sinα = ? ; sinα = ? 2

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

sinα + sinβ = 2sin α + β cos α – β , 2 2 sinα - sinβ = 2sin α - β cos α + β, 2 2cosα + cosβ = 2cos α + β cos α - β 2 2cosα - cosβ = -2sin α + β sin α - β . 2 2

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

sin75°cos75° = 2sin75°cos75°=sin(2.75°) = sin150° = 1 = 1 2 2 2.2 4sin15°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 = 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 2= 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 2 2

sin12°cos48° + cos12°sin48° = sin (12°+48°) = sin60° = √3 2

cos78° cos 18° + sin78° sin 18° = cos (78° - 18°) = cos60° = 1 2

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

2

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

2