Trigonometri

Basic Formulae
1. Sin² x + Cos² x = 1
2. 1 + tan² x = sec² x
3. 1 + cotan² x = cosec² x


Trigonometrical ratios for sum and difference
1. Sin (a+b) = sin a.cos b + cos a.sin b
2. Sin (a-b) = sin a.cos b - cos a.sin b
3. Cos (a+b) = cos a.cos b - sin a.sin b
4. Cos (a-b) = cos a.cos b + sin 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)
7. Cot (a+b) = (cot a. cot b - 1)/(cot a + cot b)
8. Cot (a-b) = (cot a. cot b + 1)/(cot a - cot b)
9. Sin (a+b).Sin (a-b) = sin^2 a - sin^2 b = Cos^2 b - cos^2 a
10. Cos (a+b).cos (a-b) = cos^2 a - sin^2 b  = cos^2 b - sin^2 a
11. Sin 2a = 2sin a. cos a   = (2 tan a)/(1+tan^2 a)
12. Cos 2a = cos^2 - sin^2 a   = 1 - 2 sin^2    = 2cos^2 a - 1  = (1- tan^2 a)/(1+ tan^2 a)
13. Tan 2a = (2tan a)/(1- tan^2 a)
14. Tan (a/2) = (1-cos a)/sin a
15. Cot (a/2) = (1+cos a)/sin a
16. Tan^2 (a/2) = (1-cos a)/(1+cos a)
17. Cot^2 (a/2) = (1+cos a)/(1- cos a)

Sum and Difference into products
1. Sin A + sin B = 2 sin 1/2(A+B) Cos 1/2 (A-B)
2. Sin A - sin B = 2 cos 1/2(A+B) Sin 1/2 (A-B)
3. Cos A + Cos B = 2 cos 1/2(A+B) Cos 1/2 (A-B)
4. Cos A - Cos B = -2 sin 1/2(A+B) sin 1/2 (A-B)
5. Tan A + Tan B = [Sin (A+B)]/(Cos A.Cos B)
6. Tan A -Tan B = [Sin (A-B)]/(Cos A.Cos B)
7. Cot A + Cot B = [Sin (B+A)/(Sin A.Sin B)]
8. Cot A - Cot B = [Sin (B-A)/(Sin A.Sin B)]

Product into Sum or Difference
1. 2Sin A.cos B = Sin (A+B) + Sin (A-B)
2. 2Cos A.Sin B = Sin (A+B) - Sin (A-B)
3. 2Cos A.Cos B = Cos (A+B) + Cos (A-B)
4. 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

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Problem Set
 1. Buktikan 2(sin^6 a+cos^6 a) - 3(sin^4 a + cos^4 a) + 1 = 0

Jawab :
Misalkan sin'^2 a = p dan cos'^2 a = q
2[(p+q)^3 - 3pq(p+q)] - 3[(p+q)'' - 2pq] + 1 = 0
2(1 - 3pq) - 3(1 - 2pq) + 1 = 0
2 - 6pq - 3 + 6pq + 1 = 0

2. Jika x bilangan real Tentukan nilai dari 3(sinx-cosx)^4 + 6(sinx+cosx)^2 + 4(sin^6 x + cos^6 x)

Jawab :
3 ((sinx - cosx)")" + 6 (sinx + cos x)" + 4 ((sin"x+cos"x)"' - 3 sin"xcos"x(sin"x+cos"x))
3 (sin"x+cos"x-2sinxcosx)" + 6(sin"x+cos"x +2sinxcosx) + 4 (1- 3sin"x.cos"x)
3 (1- 2sinxcosx)" + 6(1+2sinxcosx) + 4 (1-3sin"xcos"x)
3 (1 -4sinxcosx + 4sin"x.cos"x) + 6(1+2sinxcosx) + 4 (1-3sin"xcos"x)
3 - 12sinxcosx + 12 sin"xcos"x + 6 + 12sinxcosx + 4 - 12 sin"x.cos"x
3 +6+4 = 13

3. Buktikan Sin^6 a + cos^6 a + (3sin^2 a.cos^2 a) = 1

Jawab :
(sin^2 a + cos^2 a)^3 - 3 sin^2 a.cos^2 a(sin^2 a+cos^2 a) + (3sin^2 a.cos^2 a)
[ 1 - 3 sin^2 a.cos^2 a] + 3 sin^2 a.cos^2 a
1

4. Tentukan nilai dari
Sin^2 5 + sin^2 10 + sin^2 15+sin^2 20 + ... + sin^2 90

Jawab :
sin^2 5 + sin^2 10+ ...+ sin^2 85 + 1
sin^2 5 + sin^2 10+ ...+ sin^2 (90-10) + sin^2 (90-5) + 1
sin^2 5 + sin^2 10 + ....+ cos^2 10+ cos^2 5 + sin^2 45 + 1
8 + 1/2 +1 = 9 1/2 = 19/2

5. Tentukan nilai dari
cosec"phi/7 + cosec" 2phi/7+cosec"4phi/7

Jawab :
‎(1+ cot" phi/7)+(1+ cot" 2phi/7)+(1+ cot" 4phi/7)
3+(cot" phi/7+cot" 2phi/7+cot" 4phi/7)

Mis A=nphi/7,n=1,2,3,4,5,6,7
7A=nphi
4A=nphi-30
Tan4A=-tan3A
Tan2(2A)=-tan3A
2tan2A/(1-tan^2 2A)=(3tanA-tan^3A)/(1-3tan^2A)
[2.2tanA(1-tan^2A)]/(1-tan^2A)^2 - 4tan^2A)=(3tanA-tan^3A)/(1-3tan^2A)
Tan^6A-21tan^4A+35tan^2A-7=0
krn
tan^2 phi/7 = tan^2 6phi/7
tan^2 2phi/7 = tan^2 5phi/7
tan^2 3phi/7 = tan^2 4phi/7
jadi
tan^3 A -21tan^2A+35tanA-7=0
dengan akar2 tan^2 phi/7, tan^2 2phi/7,tan^2 4phi/7
(cot" phi/7+cot" 2phi/7+cot" 4phi/7)= (c/a)/(-d/a) = 35/7 = 5
jadi
3+(cot" phi/7+cot" 2phi/7+cot" 4phi/7) = 3+5=8

6. TEntukan nilai dari
tan"(pi/16)+ tan"(3 pi/16) +tan"(5 pi/16) + tan"(7pi/16)

Jawab :
‎7phi/16 = phi/2 - phi/16-->tan 7phi/16 = cotan phi/16
5phi/16 = phi/2 - 3phi/16-->tan 5phi/16 = cotan 3phi/16
mis phi/16 = x
tan"x+tan"3x+cotan"3x+cotan"x
dipisah
(tan"x+cotan"x)+(tan"3x+cotan"3x)

* (tan"x+cotan"x) = (tanx+cotanx)" -2
(1/sinxcosx)" - 2
(2/sin2x)" - 2
(8/(1-cos 4x) - 2 -->4x=phi/4
(8/(1-1/2V2) - 2
14 + 8V2

*(tan"3x+cotan"3x) = (tan3x+cotan3x)" - 2
(8/(1-cos 12x) - 2 -->12x = 3phi/4
(8/(1+1/2V2) - 2
14 - 8V2
(tan"x+cotan"x)+(tan"3x+cotan"3x) = 14 + 8V2 + 14 - 8V2 = 28

7. Tentukan nilai dari
sin phi/2011. sin 2phi/2011... sin 2010phi/2011Jawab :
Sin phi/m.Sin 2phi/m.Sin 3phi/m....Sin (m-1)phi/m = [m/ 2^(m-1)]
jadi
sin phi/2011. sin 2phi/2011... sin 2010phi/2011 = 2011/(2^2010)

8. Tentukan nilai dari sin phi/7. sin 2phi/7. sin 3phi/7. sin 4phi/7. sin 5phi/7. sin 6phi/7

Jawab:
Mis a=pi/7
Sina = tana.cosa
7a=pi
4a=pi-3a
Sin 4a=sin3a
Sin 5a=sin2a
(Tanacosa.tan2acos2a.tan4a.cos4a)^2
(tana.tan2a.tan4a)^2.(Cosa.cos2a.cos4a)^2
(7).(-1/8)^2 7/64

9. Tentukan nilai cari
cos 20.cos 40.cos 80
jawab :

cos 20.cos 40.cos 80
= sin (2^3 . 20)/ 2^3 sin 20= sin 160 / 8 sin 20= sin (180 - 20) / 8 sin 20= sin 20/ 8 sin 20
= 1/8

10. Diket 0<=a<=pi/2 & 0<=b<=pi/2.
jika sin a-sin b=3/5,& cos a+cos b=4/5,
maka sin (a+b)=
Jawab :
‎(sin a-sin b)^2=(3/5)^2
sin^2 a+sin^2 b-2sina.sinb=9/25....(1)
(cos a+cos b)^2=(4/5)^2
cos^2 a+cos^2 b+2cosa.cosb=16/25...(2)
Jumlahkan (1) dan (2)
2 + 2cosa.cosb-2sina.sinb=1
2(cosa.cosb-sina.sinb)=-1
(cosa.cosb-sina.sinb)=-1/2
Cos (a+b)=-1/2
Sin(a+b)=(V3)/2