TRIGNOMETERY
TRIGNOMETRY
1. Angle = arc / radius
θ = x / r
unit is radian. (rad)
2. Relation b/w degree and radian
1. 1° = π / 180 Rad.
2. 1 Rad = 180 / π degree
Smaller unit is:
1° = 60' (minute)
1' = 60" (Second)
3. R1 / R2 = θ1 / θ2 (R=radius, θ=angle)
4. Sin2x = 2 Sinx Cosx
= 2 tanx / 1 + tan²x
5. Cos2x = 2 cos²x-1
= 1 - 2 sin²x
= cos²x - sin²x
= 1 - tan²x / 1 + tan²x
6. tan2x = 2 tanx / 1 - tan²x
7. tan3x = 3 tanx - tan³x / 1 - 3 tan²x
8. Sin3x = 3 sinx - 4sin³x
9. Cos3x = 4 cos³x - 3cosx
10. Half angle formula
i) Sinx = 2 sin x/2 cos x/2 = 2 tan x/2 / 1 - tan² x/2
ii) Cosx = 2 cos² x/2–1 = 1-2sin²x/2= 1 - tan² x/2 / 1 + tan² x/2
11. Sin (x+y) = Sinx Cosy + Cosx Siny
12. Sin (x-y) = Sinx Cosy - Cosx Siny
13. Cos (x+y) = Cosx Cosy - SinxSiny
14. Cos (x-y) = Cosx Cosy + SinxSiny
15. tan (x+y) = tanx + tany / 1 - tanx tany
16. tan (x-y) = tanx - tany / 1 + tanx tany
17. Cot (x±y) = Cotx Coty ∓ 1 / Coty ± Cotx
18. Sinn+Siny = 2sin(x+y/2) . cos(x-y/2)
19. Sinn-Smy = 2 cos(x+y/2) . sin(x-y/2)
20. cosx+cosy = 2cos(x+y/2) . cos(x-y/2)
21. cosx-cosy = -2sin(x+y/2) sin(x-y/2)
22. 2sinncosy = sin(x+y) + sin(x-y)
23. 2cosx cosy = cos(x+y) + cos(x-y)
24. 2 sinnsiny = cos(x-y) - cos(x+y)
All Student Take chem.
1. sin(90-v) = cosθ
cos(90-θ) = sinθ
tan(90-θ) = cotθ
cot(90-θ) = tanθ
sec(90-θ) = cscθ
cosec(90-θ) = secθ.
2. sin(90+θ) = cosθ
cos(90+θ) = -sinθ
tan(90+θ) = -cotθ
cot(90+θ) = -tanθ
sec(90+θ) = -cscθ
cosec(90+θ) = cotθ.
3. Sin(180-θ) = Sinθ
Cos(180-θ) = -Cosθ
tan(180-θ) = -Cotθ tanθ.
Cot(180-θ) = - Cotθ.
Sec(180-θ) = - Secθ
Cosec(180-θ) = Cosecθ
4. Sin(180+θ) = - Sinθ
Cos(180+θ) = -Cosθ
tan(180+θ) = tanθ
Cot(180+θ) = Cotθ
Sec(180+θ) = -Secθ
Cosec(180+θ) = -Cosecθ.
5. Sin(270-θ) = -Cosθ
Cos(270-θ) = -Sinθ.
tan(270-θ) = Cotθ
Cot(270-θ) = tanθ
Sec(270-θ) = -Cosecθ
Cosec(270-θ) = -Secθ
6. Sin(270+θ) = -Cosθ
Cos(270+θ) = Sinθ
Tan(270+θ) = -Cotθ
Cot(270+θ) = -tanθ.
Sec(270+θ) = +Cosecθ
Cosec(270+θ) = -Secθ.
7. Sin(360-θ) = - Sinθ
Cos(360-θ) = Cosθ
Tan(360-θ) = -Tanθ
Cot(360-θ) = - Cotθ
Sec(360-θ) = Secθ
Cosec(360-θ) = -Cosecθ.
8. Sin(n X 360-θ) = - Sinθ
(n is any integer)
9. Sin(360+θ) = Sinθ
Cos(360+θ) = Cosθ
Tan(360+θ) = Tanθ
Cot(360+θ) = Cotθ
Sec(360+θ) = Secθ
Cosec(360+θ) = Cosecθ.
Note: Sin(n x 360+θ) = Sinθ
2. If we have odd integral multiple of 90° then function are
interchanged
Sinθ ↔ Cosθ Tanθ ↔ Cotθ
Secθ ↔ Cosecθ
3. If we have even integral multiple of 90, function remain same
but sign changes.
Trig. Equation.
1. If sinθ = 0 then θ = nπ, n=0,1,2,...
2. If cosθ = 0 [ θ = 2nπ ]
3. If tanθ = 0 [ θ = nπ ]
4. If sinx = siny then x = nπ+(-1)n.y
5. If cosx = cosy [ x = 2nπ ± y ]
6. If tanx = tany [ x = nπ + y ]
7. If sin²x = sin²y [ x = nπ ± y ]
8. If tan²x = tan²y [ x = nπ ± y ]
Domain Range
Sin [-π/2, π/2] [-1,1]
Cos [0, π] [-1,1]
Tanθ [-π/2, π/2] (∞, ∞)
Cot [0,π] (-∞, ∞)
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