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

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

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

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

5. ОСНОВНИ ЗАДАЧИ и функции Main tasks and functions
Sin(150°)= sin150° = tg(60°) = tg60° = √3 2
Cos(135°) = cos135° = -√ 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

6. 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

7. 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

8. 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,…

9. 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°
sin65°.cos25° + sin25°.cos65° = sin(65°+25°) = sin90° = 1

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

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

12. ∆ABC , AC=BC=10cm , sinα =4 , S =? C
Р-е : (Pythagorean theorem)
∆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
S = AB.CH = 12.8 = 48cm²
h = = 8cm 5

13. 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
cos α = ± √ 1 +cosα
tg α = ± √ 1 – cosα
cosα
cotg α = ± √ 1 + cosα
cosα

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

15. cos25°cos65°=cos(90°-65°) cos65°= 2sin65°.cos65 °
sin130° = 1 sin130°
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

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

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

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

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

20. sin75°cos75° = 2sin75°cos75°=sin(2.75°) = sin150° = 1 = 1
sin15°cos15°= 2sin15°cos15° = sin(2.15°) = sin 30° = 1 = 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 – 2sinα = 2(√2 – sinα) = 2(sin45° - sinα) =
= 2(sin45°-sinα) = 2(2sin 45°+α cos 45° - α) =
= 4 sin45°+α cos 45°- α
2sinα +√3 = 2(sinα +√3) = 2(√3 + sinα) =
= 2(sin60° + sinα) = 2(2sin 60°+α cos 60°-α) =
= 2(2sin( 30°+α ) cos( 30° - α) =
= 4sin (30°+ α) cos (30°- α)

22. cosα + cos5α - cos2α - cos4α = 2cos 6α cos 4α – (cos2α + cos4α) =
= 2cos 6α cos 4α – 2cos 6α cos(-2α) =
= 2cos 6α (4α – cos2α) = 2cos 3α (cos2α - cosα) =
= 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

23. 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

24. 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