Tabel Rumus Trigonometri

1. Pengganda Decimal

10 1	deka (da)	10 -1	deci (d)
10 2	hecto (h)	10 -2	centi (c)
10 3	kilo (k)	10 -3	milli (m)
10 6	mega (M)	10 -6	micro (u)
10 9	giga (G)	10 -9	nano (n)
10 12	tera (T)	10 -12	pico (p)
10 15	peta (P)	10 -15	femto (f)
10 18	exa (E)	10 -18	atto (a)

2. Seri.

Maclaurin Series.

1.      e ^x = 1 + x + x ² / 2! + ... + x ⁿ / n! + ...

untuk semua x

2.       sin x = x - x ³ / 3! + x ⁵ / 5! - x ⁷ / 7! + ...

untuk semua x

3.       cos x = 1 - x ² / 2! + x ⁴ / 4! - x ⁶ / 6! + ...

untuk semua x

4.       ln(1 + x) = x - x ² / 2 + x ³ / 3 -... + (-1) ⁿ⁺¹ x ⁿ / n + ...

untuk (-1 < x <= 1)

5.       tan x = x + (1/3) x ³ + (2/15) x ⁵ + (17/315) x ⁷ + ...

untuk (-Pi/2 < x < Pi/2)

6.       arcsin x = x + (1/2) x ³ / 3 + (1.3/2.4) x ⁵ / 5 + (1.3.5/2.4.6) x ⁷ / 7 + ...

untuk (-1 < x < 1)

7.       arctan x = x - x ³ / 3 + x ⁵ / 5 - ...

untuk (-1 < x < 1)

8.       sinh x = x + x ³ / 3! + x ⁵ / 5! + x ⁷ / 7! + ...

untuk semua x

9.       cosh x = x + x ² / 2! + x ⁴ / 4! + x ⁶ / 6! + ...

untuk semua x

10.       arcsinh x = x - (1/2) x ³ / 3 + (1.3/2.4) x ⁵ / 5 - (1.3.5/2.4.6) x ⁷ / 7 + ...

untuk (-1 < x < 1)

11.       1 / (1 - x) = 1 + x + x ² + x ³ + ...

untuk (-1 < x < 1)

Arithmetic Series.

12.       Sn = a + (a + d) + (a + 2d)+...+(a + [n-1]d)
= (n/2)[first term + last term]
= (n/2)[a + (a+[n - 1]d)
= n(a + [n - 1]d)

Seri Geometris.

13.       Sn = a + a r + a r ² + a r ³ +...+ a r ^n-1 = a (1 - r ⁿ)/(1 - r)

Integer Series.

14.       1 + 2 + 3 + ... + n = (1 / 2) n (n + 1)
15.       1 ² + 2 ² + 3 ² + ... + n ² = (1 / 6) n (n + 1)(2n + 1)
16.       1 ³ + 2 ³ + 3 ³ + ... + n ³ = [ (1 / 2) n (n + 1) ] ²

3. faktorial, Permutasi dan Kombinasi.

1.       n factorial = n ! = n.(n-1).(n-2)...2.1
2.      Permuatations dari n objek yang diambil r pada saat itu:
         n P r = n ! / [ (n - r) ! ]
3.      Kombinasi dari n objek yang diambil r pada saat itu:
         n C r = n ! / [ r ! (n - r) ! ]

4. Ekspansi Binomial (Formula).

1. Jika n adalah bilangan bulat positif, kita dapat memperluas (x + y) ⁿ sebagai berikut
(X + y) ⁿ = n C 0 x ⁿ + n C 1 x ^{n - 1} y + n C 2 x ^{n - 2} y ² + ... + n C ny ⁿ
Istilah umum n C r diberikan oleh

n C r = n! / [R! (N - r)! ]

5. Rumus trigonometri.

Sum / Perbedaan Rumus Angles.

1. cos (A + B) = cos A cos B - dosa A sin B
2. cos (A - B) = cos A cos B + sin sin A B
3. sin (A + B) = sin A cos B + cos A sin B
4. sin (A - B) = sin A cos B - cos A sin B
5. tan (A + B) = [tan A + tan B] / [1 - tan A tan B]
6. tan (A - B) = [tan A - tan B] / [1 + tan A tan B]

Sum / Perbedaan Rumus Fungsi trigonometri.

7. sin A + sin B = 2 sin [(A + B) / 2] cos [(A - B) / 2]
8. sin A - sin B = 2 cos [(A + B) / 2] sin [(A - B) / 2]
9. cos A + cos B = 2 cos [(A + B) / 2] cos [(A - B) / 2]
10. cos A - B = cos - 2 sin [(A + B) / 2] sin [(A - B) / 2]

Produk rumus fungsi trigonometri.

11. 2 sin A cos B = sin (A + B) + sin (A - B)
12. 2 cos A sin B = sin (A + B) - sin (A - B)
13. 2 cos A cos B = cos (A + B) + cos (A - B)
14. 2 sin A sin B = - cos (A + B) + cos (A - B)

Beberapa Rumus rangkap.

15. sin 2A = 2 sin A cos A
16. cos 2A = cos ² A - sin ² A = 2 cos ² A - 1 = 1-2 sin ² A
17. sin 3A = 3 sin A - 4 sin ³ A
18. cos 3A = 4 cos ³ A - 3 cos A

Daya Mengurangi Rumus.

19. sin ² A = (1/2) [1 - cos 2A]
20. cos ² A = (1/2) [1 + cos 2A]
Fungsi hiperbola
sinh² x = ½cosh 2x — ½
cosh² x = ½cosh 2x + ½
sinh³ x = ¼sinh 3x — ¾sinh x
cosh³ x = ¼cosh 3x + ¾cosh x
sinh⁴ x = 3/8 - ½cosh 2x + 1/8cosh 4x
cosh⁴ x = 3/8 + ½cosh 2x + 1/8cosh 4x

Basic Formulae
21. Sin² x + Cos² x = 1
22. 1 + tan² x = sec² x
23. 1 + cotan² x = cosec² x

Trigonometrical ratios for sum and difference
24. Sin (a+b) = sin a.cos b + cos a.sin b
25. Sin (a-b) = sin a.cos b - cos a.sin b
26. Cos (a+b) = cos a.cos b - sin a.sin b
27. Cos (a-b) = cos a.cos b + sin a.sin b
28. Tan (a+b) = (tan a + tan b)/(1- tan a.tan b)
29. Tan (a-b) = (tan a - tan b)/(1+ tan a.tan b)
30. Cot (a+b) = (cot a. cot b - 1)/(cot a + cot b)
31. Cot (a-b) = (cot a. cot b + 1)/(cot a - cot b)
32. Sin (a+b).Sin (a-b) = sin^2 a - sin^2 b = Cos^2 b - cos^2 a
33. Cos (a+b).cos (a-b) = cos^2 a - sin^2 b  = cos^2 b - sin^2 a
34. Sin 2a = 2sin a. cos a   = (2 tan a)/(1+tan^2 a)
35. Cos 2a = cos^2 - sin^2 a   = 1 - 2 sin^2    = 2cos^2 a - 1  = (1- tan^2 a)/(1+ tan^2 a)
36. Tan 2a = (2tan a)/(1- tan^2 a)
37. Tan (a/2) = (1-cos a)/sin a
38. Cot (a/2) = (1+cos a)/sin a
39. Tan^2 (a/2) = (1-cos a)/(1+cos a)
40. Cot^2 (a/2) = (1+cos a)/(1- cos a)

Sum and Difference into products
41. Sin A + sin B = 2 sin 1/2(A+B) Cos 1/2 (A-B)
42. Sin A - sin B = 2 cos 1/2(A+B) Sin 1/2 (A-B)
43. Cos A + Cos B = 2 cos 1/2(A+B) Cos 1/2 (A-B)
44. Cos A - Cos B = -2 sin 1/2(A+B) sin 1/2 (A-B)
45. Tan A + Tan B = [Sin (A+B)]/(Cos A.Cos B)
46. Tan A -Tan B = [Sin (A-B)]/(Cos A.Cos B)
47. Cot A + Cot B = [Sin (B+A)/(Sin A.Sin B)]
48. Cot A - Cot B = [Sin (B-A)/(Sin A.Sin B)]

Product into Sum or Difference
49. 2Sin A.cos B = Sin (A+B) + Sin (A-B)
50. 2Cos A.Sin B = Sin (A+B) - Sin (A-B)
51. 2Cos A.Cos B = Cos (A+B) + Cos (A-B)
52. 2Sin A. Sin B = -Cos (A+B) + Cos (A-B)

Triple Angles
 Sin (a+b+c) = Sin a Cos b Cos c + Cos a Sin b Cos c + Cos a Cos b Sinc + Sin a Sin b Sin c
Cos (a+b+c) =Cos a Cos b Cos c - cos a Sin b Sin c - Sin a Cos b Sin c - Sin a Sin b Cos c
Tan (a+b+c) = Sin (a+b+c) / Cos (a+b+c)
Sin 3a = 3.Sin a - 4 Sin^3 a
Cos 3a = 4.Cos^3 a - 3.Cos a
Tan 3a = (3 Tan a - Tan^3 a)/(1 - 3.Tan^2 a)
Sin a.Sin (60-a).Sin (60+a) = (Sin 3a)/4
Cos a.Cos(60-a).Cos(60+a) = (Cos 3a)/4

Product dan Sum of the sin and cosine Series
Cos a. Cos 2a. Cos 4a. Cos 8a....Cos 2^(n-1) = [Sin ((2^n).a)] / (2^n .Sin a)
Sin a + Sin (a+b) + Sin (a+2b) + ... n  terms = Sin [a+(n-1)b]. [Sin (n .b/2)]/(sin (b/2))
Cos a+ Cos (a+b) + Cos (a+2b) + ... n  terms = Cos [a+(n-1)b]. [Sin (n .b/2)]/(sin (b/2))
Sin phi/m.Sin 2phi/m.Sin 3phi/m....Sin (m-1)phi/m = [m/ 2^(m-1)]
1/(sina.sin2a) +1/(sin2a.sin3a) +...+ 1/(sin an.sin(n+1)a)=[cot a - cot (n+1)a]/sin a
1/(cos a-cos3a)+1/(cos a-cos5a)+...+1/(cos a-cos(2n+1)a = [cot a-cot (n+1)a]/2sina
1/(cos a+cos3a)+1/(cos a+cos5a)+...+1/(cos a+cos(2n+1)a =[tan(n+1)a- tan a]/2sina

sec(x) = 1/cos(x),

csc(x) = 1/sin(x),

cot(x) = 1/tan(x),

tan(x) = sin(x)/cos(x),

cot(x) = cos(x)/sin(x).

sin(-x) = -sin(x),

cos(-x) = cos(x),

tan(-x) = -tan(x),

cot(-x) = -cot(x),

sec(-x) = sec(x),

csc(-x) = -csc(x).

sin(/2-x) = cos(x),

cos(/2-x) = sin(x),

tan(/2-x) = cot(x),

cot(/2-x) = tan(x),

sec(/2-x) = csc(x),

csc(/2-x) = sec(x).

sin(/2+x) = cos(x),

cos(/2+x) = -sin(x),

tan(/2+x) = -cot(x),

cot(/2+x) = -tan(x),

sec(/2+x) = -csc(x),

csc(/2+x) = sec(x).

sin(-x) = sin(x),

cos(-x) = -cos(x),

tan(-x) = -tan(x),

cot(-x) = -cot(x),

sec(-x) = -sec(x),

csc(-x) = csc(x).

sin(+x) = -sin(x),

cos(+x) = -cos(x),

tan(+x) = tan(x),

cot(+x) = cot(x),

sec(+x) = -sec(x),

csc(+x) = -csc(x).

sin(2+x) = sin(x),

cos(2+x) = cos(x),

tan(2+x) = tan(x),

cot(2+x) = cot(x),

sec(2+x) = sec(x),

csc(2+x) = csc(x).

sin2(x) + cos2(x) = 1,

tan2(x) + 1 = sec2(x),

1 + cot2(x) = csc2(x).

sin(x+y) = sin(x)cos(y) + cos(x)sin(y),

cos(x+y) = cos(x)cos(y) - sin(x)sin(y),

tan(x+y) = [tan(x)+tan(y)]/[1-tan(x)tan(y)],

cot(x+y) = [cot(x)cot(y)-1]/[cot(x)+cot(y)].

sin(x-y) = sin(x)cos(y) - cos(x)sin(y),

cos(x-y) = cos(x)cos(y) + sin(x)sin(y),

tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)],

cot(x-y) = [cot(x)cot(y)+1]/[cot(y)-cot(x)].

sin(2x) = 2 sin(x)cos(x),

cos(2x) = cos2(x) - sin2(x),

            = 2 cos2(x) - 1,

            = 1 - 2 sin2(x),

tan(2x) = [2 tan(x)]/[1-tan2(x)],

cot(2x) = [cot2(x)-1]/[2 cot(x)].

|sin(x/2)| = sqrt([1-cos(x)]/2),

 |cos(x/2)| = sqrt([1+cos(x)]/2),  

 |tan(x/2)| = sqrt([1-cos(x)]/[1+cos(x)]),

 tan(x/2) = [1-cos(x)]/sin(x),

            = sin(x)/[1+cos(x)] 

sin(x)cos(y) = [sin(x+y) + sin(x-y)]/2,

cos(x)sin(y) = [sin(x+y) - sin(x-y)]/2,

cos(x)cos(y) = [cos(x-y) + cos(x+y)]/2,

sin(x)sin(y) = [cos(x-y) - cos(x+y)]/2.

sin(x) + sin(y) = 2 sin[(x+y)/2]cos[(x-y)/2],

sin(x) - sin(y) = 2 cos[(x+y)/2]sin[(x-y)/2],

cos(x) + cos(y) = 2 cos[(x+y)/2]cos[(x-y)/2],

cos(x) - cos(y) = -2 sin[(x+y)/2]sin[(x-y)/2],

tan(x) + tan(y) = sin(x+y)/[cos(x)cos(y)],

tan(x) - tan(y) = sin(x-y)/[cos(x)cos(y)],

cot(x) + cot(y) = sin(x+y)/[sin(x)sin(y)],

cot(x) - cot(y) = -sin(x-y)/[sin(x)sin(y)].

[sin(x)+sin(y)]/[cos(x)+cos(y)] = tan[(x+y)/2],

[sin(x)-sin(y)]/[cos(x)+cos(y)] = tan[(x-y)/2],

[sin(x)+sin(y)]/[cos(x)-cos(y)] = -cot[(x-y)/2],

[sin(x)-sin(y)]/[cos(x)-cos(y)] = -cot[(x+y)/2],

[sin(x)+sin(y)]/[sin(x)-sin(y)] = tan[(x+y)/2]/tan[(x-y)/2]

cotan (α + β) = (cot α.cot β - 1)/ (cot α + cot β)
cotan (α - β) = (cot α.cot β + 1)/ (cot β - cot α)
cosec (α + β) = (sec α.csc α.sec β.csc β) / (sec αcsc β + csc α.sec β)
cosec (α - β) = (sec α.csc α.sec β.csc β) / (sec αcsc β - csc α.sec β)
sec  (α + β) = (sec α.csc α.sec β.csc β) / (csc α.csc β + sec α.sec β)
sec (α - β) = (sec α.csc α.sec β.csc β) / (csc α.csc β - sec α.sec β)
sec (α + β) = (sec α.sec β) / (1 - tan α.tan β)
sec (α - β) = (sec α.sec β) / (1 + tan α.tan β)
cosec (α + β) = (cosec α.cosec β)/(cotan α + cotan β)
cosec (α - β) = (cosec α.cosec β)/(cotan α - cotan β)
Tan (a+b+c) = {tan a +tan b + tan c – tan a tan b tan c}/{1-tan a tan b – tan b tan c – tan c tan a}

Identitas Sinus

1.      sin α = ± {√1-cos²α}

2.      sin α = ± tan α/{√1+tan²α}

3.      sin α = 1/csc α

4.      sin α = ± {√sec²α – 1}/sec α

5.      sin α = ± 1/{√1+cot²α}

Identitas cosinus

1.    cos α = ±{√1-sin²α}

2.    cos α = ±1/{√1+tan²α}

3.    cos α = 1/sec α

4.    cos α = ± {√csc²α-1}/csc α

5.    cos α = ± {cot α}/{√1+cot²α}

Identitas tangent

1.      tan α = ±{√sec²α-1}

2.      tan α = ± 1/{√csc²α-1}

3.      tan α = 1/cot α

4.      tan α = ± sin α/{√1-sin²α}

5.      tan α = ± {√1-cos²α}/cos α

Identitas cosecant

1.      csc α = ± {√1+cot²α}

2.      csc α = ± 1/{√1-cos²α}

3.      csc α = 1/sin α

4.      csc α = ± {√1+tan²α}/tan α

5.      csc α = ± sec α/{√sec²α-1}

Identitas secant

1.      sec α = ± {√1+tan²α}

2.      sec α = ± 1/ {√1-sin²α}

3.      sec α = ± 1/cos α

4.      sec α = ± {√1+cot²α}/cot α

5.      sec α = ± csc α/{√csc²α-1}

Identitas cotangent

1.      cot α = ±{√csc²α-1}

2.      cot α = ± 1/{√sec²α-1}

3.      cot α = 1/tan α

4.      cot α = ±{√1-sin²α}/sin α

5.      cot α = ± cos α/{√1-cos²α}

aturan sinus

sin α/A = sin β/B = sin γ/C

aturan cosinus

a² = b² + c² - 2bc cos α → cos α = { b² + c² - a²}/2bc

b² = a² + c² - 2ac cos β → cos β = {a² + c² - b²}/2ac

c² = a² + b² - 2ab cos γ → cos γ = {a² + b² - c²}/2ab 

aturan tangent

[a-b]/[a+b] = [tan{½(α-β)}]/[tan{½(α+β)}]

[b-c]/[b+c] = [tan{½(β-γ)}]/[tan{½( β-γ)}]

[a-c]/[a+c] = [tan{½(α-γ)}]/[tan{½(α+γ)}]

Formula moolweid’s

[a+b]/c = [cos{½(α-β)}]/ [sin{½}γ]

1.      Pembuktian bahwa 1+cot²α = cosec²α,

1+(cos²α/ sin²α) = cosec²α,

(sin²α/sin²α)+(cos²α/ sin²α) = cosec²α,

(sin²α+cos²α)/ (sin²α) = cosec²α,

1/ sin²α = cosec²α

2.       Pembuktian bahwa cot α = 1/tan α

Tan α = sin α/cos α sedangkan cot α = cos α/sin α, jadi

= 1/{sin α/cos α} = 1/tan α

3.       Pembuktian bahwa 1 + tan²α = sec²α,

cos²α + sin²α = 1,

{cos²α + sin²α}/ cos²α = 1/ cos²α,

(cos²α/cos²α)+(sin²α/ cos²α) = 1/ cos²α,

 {1 + tan²α} = sec²α

Tan (a+b+c) = tan {(a+b)+c} = {tan (a+b) + tan c}/{1 - tan(a+b)tanc} = {(tan a + tan b/1 – tan a.tan b) + tan c}/{1 – (tan a + tan b/1 – tan a.tan b)tan c} = {(tana + tanb + tanc – tana.tanb.tanc)/(1 – tan a.tan b)}/ {(1 – tan a.tan b – tana.tanc – tanb.tanc)/(1 – tan a.tan b)},

Tan (a+b+c) = (tana + tanb +tan c - tana tanb tanc)/ (1 – tan a.tan b – tana.tanc – tanb.tanc)
Cot (α+β+ς) = {Re (i+cot α)(i+cot β)(i+cot c)}/{Im (i+cot α)(i+cot β)(i+cot c)}
            = {cotα.cotβ.cotς – cotα – cotβ – cotς} / {cotα.cotβ + cotα.cotς + cotβ.cot ς – 1}