sin x/1+cos x = a.1+cos x/sin x b.1-cos x/sin x c.sin x/1-cos x d.sin x/cos x-1 e.1-cos x/cos x

1. sin x/1+cos x = a.1+cos x/sin x b.1-cos x/sin x c.sin x/1-cos x d.sin x/cos x-1 e.1-cos x/cos x

jawab B
sin x = √( 1 - cos² x) = √(1+cos x). √(1-cos x)
.
sin x / (1 + cos x) = √(1+cos x). √(1-cos x) / √(1+cos x).√(1+cos x)
sin x / (1 + cos x) = √(1-cos x)/√(1+cos x)
= √(1-cos x). √(1-cos x) / √(1+cos x) .√(1-cos x)
= (1- cos x) / (√(1 - cos² x)
= (1- cos x)/ (sin x)

2. (1 + cos x)² - (1 - cos x)² = 4 cos x

Uraikan.
[1+2.cos(x)+cos²x] - [1-2.cos(x)+cos²x]
= 1 - 1 + 2.cos(x)+2.cos(x) + cos²x - cos²x
= 4.cos(x)
Ruas Kiri dapat diubah menjadi:
[tex][(1+cosx)-(1-cosx)][(1+cosx)(1-cosx)]=(2cosx)(2)=4cosx[/tex]

3. 1-Sin X/Cos X = A. Cos X/1+Sin XB. Cos X/1-Sin XC. Sin X/1+Cos XD. SinX/1-Cos XE. Cos X​

Penjelasan dengan langkah-langkah:

pembahasan pada gambar

4. Tunjukkan bahwa cos 2X-cos x/cos x-1=2cos x+1

(cos 2x - cos x) / (cos x - 1)
= (2 cos² x - 1 - cos x) / (cos x - 1)
= (2 cos x + 1)(cos x - 1) / (cos x - 1)
= 2 cos x + 1

TerBukTi

5. Identitas untuk sinx / (1+cosx)= a.1-cos x/sin x b.1+cos x/sin x c.sin x/1-cos x d.sinx/cos x -1 e.1-cos x/cos x

sinx/(1+cosx)

= sinx / (1+cosx) * (1-cosx)/(1-cosx)
= (sinx)(1-cosx) / (1 - cos²x)
= (sinx)(1-cosx) / (sin²x)
= (1-cosx)/sinx
= A.

6. (Cos x-1)(cos x+ 1) =....

(cosdikali-1)=galat (cosdikali+1)=galat

7. Buktikan !sin x - cos x + 1        sin x + 1______________  =  _________sin x + cos x -1            cos x

sin x - cos x = -1
1 = sin x + cos x
sin x + 1 = cos x
jadi tinggal di bolak balik
sin x - cos x diganti jadi 1
+1 dijadiin sin x + cos x

8. Jika 0o < x < 90o maka jumlah dari cos x + cos x sin x + cos x sin2 x+ cos x sin3 x + cos x sin4 x + …. sama dengan … (A) sin x / 1 - cos x (B) cos x/ 1 + sin x (C) cos x / 1 - cos x (D) 1 + cos x / sin x (E) sec x + tan x

jawab:

(A) sin x / (1 - cos x)

9. Buktikan : (1 + cos x)(1 - cos x) = sin x cos x tan x

Jawab:

terbukti

Penjelasan dengan langkah-langkah:

perhatikan bahwa tan x = sin x/cos x maka

(1 + cos x)(1 - cos x) = sin x cos x tan x

(1² - cos²x) = sin x cos x (sin x/cos x)

gunakan sifat sin² x + cos² x = 1, maka

(1 - cos² x) = sin x cos x (sin x/cos x)

(sin² x + cos² x - cos² x) = (sin x)²

sin² x = sin² x

karena sisi kanan dan kiri sama, maka terbukti bahwa

(1 + cos x)(1 - cos x) = sin x cos x tan x

10. cos 1/2pi x cos 1/3pi x cos 1/6pi =

cos1/2pi x cos1/3pi x cos1/6pi
cos(1/2 x 180° ) x cos(1/3 x 180° ) x cos(1/6 x 180°)
cos90° x cos60° x cos30°
0 x 1/2 x 1/2√3
0

11. Buktikan !sin x - cos x + 1        sin x + 1______________  =  _________sin x + cos x -1            cos x

[tex]\dfrac{\sin x-\cos x+1}{\sin x+\cos x-1}=\dfrac{\sin x-\cos x+1}{\sin x+\cos x-1}\cdot \dfrac{\sin x+\cos x+1}{\sin x+\cos x+1}[/tex]

[tex]\dfrac{\sin x-\cos x+1}{\sin x+\cos x-1}=\dfrac{\left( \sin x+1 \right)-\cos x}{\left( \sin x+\cos x \right)-1}\cdot \dfrac{\left( \sin x+1 \right)+\cos x}{\left( \sin x+\cos x \right)+1}[/tex]

[tex]\dfrac{\sin x-\cos x+1}{\sin x+\cos x-1}=\dfrac{{{\left( \sin x+1 \right)}^2}-\cos^2 x}{{{\left( \sin x+\cos x \right)}^2}-{1^2}}[/tex]

[tex]\dfrac{\sin x-\cos x+1}{\sin x+\cos x-1}=\dfrac{\sin^2x+2\sin x+1-\cos^2x}{\sin^2x+2\sin x\cos x+\cos^2x-1}[/tex]

[tex]\dfrac{\sin x-\cos x+1}{\sin x+\cos x-1}=\dfrac{\sin^2x+2\sin x+\sin^2x}{1+2\sin x\cos x-1}[/tex]

[tex]\dfrac{\sin x-\cos x+1}{\sin x+\cos x-1}=\dfrac{2\sin^2x+2\sin x}{2\sin x\cos x}[/tex]

[tex]\dfrac{\sin x-\cos x+1}{\sin x+\cos x-1}=\dfrac{\sin x+1}{\cos x}[/tex]

12. Buktikan Identitas Trigonometri berikut 1. cos^4x - sin^4x (per) 1 - tan x = cos^2 x + sin x + cos x 2. sin^2x (per) 1 - cos x = 1 + cos x 3. 1 - cos ^4x (per) 1 + cos^2x = sin^2x

pembuktiannya terlampir ya,have a nice weekend:)

13. buktikan trigonometri (1 + cos X)² - (1 - cos X)² = 4 Cos X

(1 + cos x)² - (1 - cos x)² = 4 Cos x

bukti:
(1 + cos x)² - (1 - cos x)²
= (1 + 2 cos x + cos² x) - (1 - 2 cos x + cos² x)
= 1 + 2 cos x + cos² x - 1 + 2 cos x - cos² x
= 2 cos x + 2 cos x
= 4 cos x (terbukti)

semoga membantu ya :)
[tex](1 + cos x)^2 - (1 - cos x)^2 = 4 cos x[/tex]
[tex](1 + cos x + 1 - cos x) (1 + cos x - 1 + cos x) = 4 cos x[/tex]
[tex](2) (2 cos x) = 4 cos x[/tex]
[tex]4 cos x = 4 cos x[/tex]
TERBUKTI

Semoga membantu :)

14. (1+sin x- cos x)²-2(1+sin x -cos x)=-2sin x .cos x​

Penjelasan dengan langkah-langkah:

ini pertanyaannya apa?

15. Buktikan identitas trigonometri berikut! √[(1 - cos x) / (1 + cos x)] = sin x / (1 + cos x)

√ [ (1 - cos x) / (1 + cos x) kali (1 + cos x)/ (1 + cos x) ]
√[ 1 - Cos²x / (1 + Cosx)² ]
√(sin²x ) / ( 1 + Cosx)²]
Sinx / 1 + Cosx
Terbukti

16. Hasil operasi tan²x + 1 + tan x sec x dapat disederhanakan menjadi.... a. 1 + cos²x / sin x b. 1 + sin²x / cos x c. 1 + cos²x / sin²x d. 1 + cos²x / cos²x e. 1 + sin²x / cos²x

Jawaban nya gk ada di pilihan .. maaf kalau saya slah..[tex]$\begin{align} tan^2 x + 1 + tan \ x \ sec \ x &= sec^2 x + tan \ x \ sec \ x \\ &= sec \ x (sec \ x + tan \ x) \\ &= \frac{1}{cos \ x}( \frac{1}{cos \ x} + \frac{sin \ x}{cos \ x} ) \\ &= \frac{1}{cos \ x}( \frac{1 + sin \ x}{cos \ x} ) \\ &= \frac{1 + sin \ x}{cos^2x} \end{align}[/tex]

17. Buktikan (1 + sin x - cos x) / (1 + sin x + cos x) + (1 + sin x + cos x) / (1 + sin x - cos x) = 2 cosec x

.

a=six-cosx

a²=six²-2sinxco+cos²x=1-2sinxcosx

b=sinx+cosx

b²=six²+2sinxc+cos²x=1+2sinxcosx

a²+b²=2

ab=sin²x-cos²x

a+b=2sinx

(1 + sin x - cos x) / (1 + sin x + cos x) + (1 + sin x + cos x) / (1 + sin x - cos x) =.

(1 + a) / (1 + b) + (1 + b) / (1 + a) =.

{(1 + a) (1+a) + (1 + b)(1+b) }/{ (1 + a)(1+b) }=.

{(1 +2a+ a²) + (1 + 2b+b²) }/{ (1 + a+b+ab)}=.

{(2+2.2sinx+ 2) }/{ (1 + 2sinx+.sin²x-cos²x )}=.

{(4+4sinx) }/{ (1 + 2sinx+.2sin²x-1 )}=.

{4(1+sinx) }/{ ( 2sinx(1+.sinx )}=.

2/( sinx}=.2 cosecx

±+++++((((

18. (cos x + 1) (cos x - 1) adalah?

(cos x + 1)(cos x - 1)
= cos² x - cos x + cos x - 1
= cos² x - 1
= sin² x(cos x + 1) (cos x - 1)
= cos² x - cos x + cos x - 1
= cos² x - 1
Mengingat identitas trigonometri cos² x + sin² x = 1, maka
= cos² x - 1
= sin² x
Identitas trigonometri sin x = +- √[(1 - cos 2x) / 2], maka
= 1/2 (1 - cos 2x)

19. Cos 2x-cos x/cos x-1=2cos x+1

(cos 2x - cos x)/(cos x - 1) = (2 cos²x - 1 - cos x)/(cos x - 1)
                                           = (2 cos x + 1)(cos x - 1)/(cos x - 1)
                                           = 2 cos x + 1

20. ( cos x +1)( cos x-1 )=-sin2 x

( cos x +1)( cos x-1 )=-sin2 x
cos2 x -1 = - sin2 x
cos2 x + sin2 x -1 =0
( humus identitas cos2 x + sin2 x = 1)
1-1=0
terbukti= (cos x + 1)(cos x -1)
= cos² x - 1
= (1 - sin² x ) - 1
= - sin² x

21. 1. cotan x - cos x/cotan x = cos² x / 1+sin x2. cosec x + tan x + cotan x = cos x +1/sin x cos x3. tan x/1-tan² x= sin x cos x/ cos² x- sin² x4.sin² x/1-cos x=1 + cos x5. (1+ cos x)² - (1- cos x)² = 4 cos x bantuin membuktikan identitas trigonometri  diatas , saya masih kesulitan.

2. = ( \frac{1}{sin x} + \frac{cos x }{sin x} ) + \frac{sin x}{cos x} [/tex] 
= \frac{1 + cos x }{sin x} + \frac{sin x}{cos x} [/tex] 
 = \frac{(1 + cos x ) cos x + sin^2x}{sin x cos x} [/tex]
=  \frac{cos x + cos^2 + sin^2 x}{sin x cos x} [/tex] 
= \frac{cos x +1}{sin x cos x} [/tex]

22. Nilai (1 + cos x) (1-cos x) sama dengan

(1 + cos x)(1 - cos x)
= 1 - cos x + cos x - cos^2 x
= 1 - cos^2 x
= sin^2 x

23. 1/1+Cos x + 1/1-Cos x

[tex] \frac{1}{1 + \cos(x) } + \frac{1}{1 - \cos(x) } = \\ \frac{1 - \cos(x) + 1 + \cos(x) }{(1 + \cos(x)) \: \times (1 - \cos(x)) } \\ \frac{2}{1 - ({ \cos(x) })^{2} } = \frac{2}{ ({ \sin(x) })^{2} } = 2 \times ({ \csc(x)})^{2} [/tex]
ini jawabannya
semoga membantu
like ya

24. buktikan bahwa 1+cos x / sin x + sin x / cos x = cos x + 1 / sin x cos x​

Jawab:

Penjelasan dengan langkah-langkah:

(1 + cos x)/sin x + sin x/cos x

= ((1 + cos x) . cos x) + sin x . sin x)/(sin x . cos x)

= (cos x + cos² x + sin² x)/(sin x . cos x)

= (cos x + 1)/(sin x . cos x)

Detail Jawaban

Kelas  10

Mapel 2 - Matematika

Bab  7 - Trigonometri

Kode Kategorisasi :  10.2.7

[tex] \frac{1 + \cos(x) }{ \sin(x) } + \frac{ \sin(x) }{ \cos(x) } [/tex]

[tex] = \frac{ \cos(x)(1 + \cos(x)) + \sin(x) ( \sin(x)) }{ \sin(x). \cos(x) } [/tex]

[tex] = \frac{ \cos(x) + \cos {}^{2} (x) + \sin {}^{2} (x) }{ \sin(x). \cos(x) } [/tex]

[tex] = \frac{ \cos(x) + 1}{ \sin(x). \cos(x) } [/tex]

[Terbukti]

25. buktikan cos⁴ x (1 - tan⁴ x) = cos² x -cos⁴ x (1 - tan⁴ x) = cos² x -sin² x² x

cos⁴(x) (1 - tan⁴ (x) ) = cos²(x) -sin²(x) <= Buktikan

cos⁴(x) (1 - tan⁴ (x) ) = cos⁴(x) (1-tan²(x))(1+tan²(x))

                                = cos⁴(x).sec²(x) .(1-tan²(x)) <= sec = 1/cos

                                 = cos²(x) . (1-tan²(x))

                                 = cos²(x) - cos²(x).tan²(x)  <= tan = sin/cos

cos⁴(x) (1 - tan⁴ (x) )   = cos²(x) - sin²(x)

                               

26. Cos x . Csc x . Tan x Cos x . Cot x + sin x Sin x : 1+cos x + sin x : 1-cos x

[tex]cos x.csc x.tan x = cos x. \frac{1}{sin x}. \frac{sinx}{cos x} = \frac{cos x}{cos x} = 1 [/tex]

[tex]cos x.cot x+sin x = cos x. \frac{1}{tan x} +sin x = cosx . \frac{cos x}{sin x} + sin x [/tex]
[tex]\frac{cos^{2}x}{sin x} = -sin x[/tex]
[tex]cos^{2}x=-sinx.sinx = -sin^{2}x[/tex]

27. Buktikan identitas berikut: cos³x + sin³x + cos²x (1+sin x) + sin²x (1-cos x) - cos x - sin x =1

cos³x + sin³x + cos²x (1+ sin x) + sin²x (1- cos x) - cos x - sin x = 1
-----------------------
cos²x + sin²x = 1
-----------------------
cos³x + sin³x + cos²x + cos²x sin x + sin²x - sin²x cos x - cos x - sin x
cos³x + sin³x + (cos²x + sin²x) + cos²x sin x + sin²x cos x - cos x - sin x
cos³x + sin³x + (1) + cos²x sin x + sin²x cos x - cos x - sin x
cos³x + sin²x cos x  + sin³x + cos²x sin x  - cos x - sin x + 1
(cos²x + sin²x) cos x + (sin²x + cos²x) sin x - cos x - sin x + 1
(1) cos x +(1) sin x - cos x - sin x + 1
cos x - cos x + sin x - sin x + 1
0 + 0 + 1 = 1

28. Tentukan x, sin x / 1-cos x = 1+cos x / cos 1/2 x

Penjelasan dengan langkah-langkah:

sinx/(1 - cosx) = (1 + cosx)/cos(1/2x)

sinx.cos1/2x = (1 - cosx)(1 + cosx)

(2sinx . cos1/2x = 1² - cos²x

(2sinx . cos1/2x = 1 - cos²x

(2sinx . cos1/2x = sin²x

2cos1/2x = sinx

2cos1/2x = 2sin1/2x cos1/2x

sin1/2x = 1

sin1/2x = sin90º

1/2x = 90º

x = 180º

catatan :

sin2x = 2sinx.cosx

sinx = 2sin1/2x.cos1/2x

′′ maaf kalau salah ′′

29. cos(x-y) = -1,maka cos x + cos y =.....

cos (x - y) = -1
cos (x - y ) = cos 180
x - y = 180
x = 180 + y (subsitusikan ke cos x + cos y
= cos (180 + y ) + cos y
= - cos y + cos y
= 0

30. Buktikan indetitas trigonometri berikut! 7.) cos x per sin x - 1 per cos x sin x = -tan x 8.) (1+cos x) pngkt 2 - (1- cos x)pngkt 2 = 4 cos x 9.) (cos x + 1) (cos x - 1) = -sin pngkt 2 x 10.) 1 - cos x per sin x = sin x per 1 + cos x pkek jalan yah! :)

semangat yaaa..belajar trigonometri harus kuat hafalannya..
semoga membantu..

31. Turunan dari f(x) = (1 + cos²x) adalah....A. 7 sinx cos x (1 + cos²x)^6B. 14 sinx cos x (1 + cos²x)^6C. -7 sin2x (1 + cos²x)^6D. 7 sin2x (1 + cos²x)^6E. -7 sinx cos x (1 + cos²x)^6​

Jawaban:

C. -7 sin 2x (1 + cos²x)⁶

Penjelasan dengan langkah-langkah:

f(x) = (1 + cos²x)⁷

u(x) = (1 + cos²x) ⇒ u'(x) = -2sin x cos x

n = 7

f '(x) = 7(1 + cos²x)⁷-¹ . -2sin x cos x

f '(x) = -7 (1 + cos²x)⁶ . -sin 2x

f '(x) = -7 sin 2x (1 + cos²x)⁶

32. Buktikan bahwa sin x - cos x + 1 / sin x + cos x - 1 = sin x / cos x

Jawab:

Pembuktian terlampir di gambar.

33. Buktikan 1 per 1+cos x +1 per 1-cos x =2 cos ec ²x

Penjelasan bisa lihat digambar

34. Bila sin x + cos x = 1/3 maka cos x / 1-sin x + sin x/ 1-cos x

cos x / 1-sinx + sinx / 1-cosx

cosx(1-cosx) + sinx(1-sinx) / (1-sinx)(1-cosx)

cosx-cos²x + sinx - sin²x / (1-cosx-sinx+sinxcosx

cosx+sinx -(cos²x+sin²x) / 1-(cosx+sinx) +sinxcosx

1/3 -(1) / 1-(cosx+sinx) +sinxcosx

-2/3 / 1-(1/3)+(-4/9)
-2/3 bagi 27-9 - 12 /27
-2/3 bagi 6/27
-2/3 kali 27/6
(-2)(27) / (3)(6)
-54/18
-3
jadi hasilnya adalah -3

hasil dari sinxcosx di bawah ini
↓
↓
sinx + cosx =1/3
(sinx+cosx)²=(1/3)²
sin²x+2sinxcosx+cos²x=1/9
1+2sinxcosx =1/9
2sinxcosx =1/9 - 1
sinxcosx= -8/9 x 1/2
sinxcosx = -4/9

35. (cos x - 1) (cos x +1) = ......

(cos x - 1) (cos x + 1) = cos^2 x - 1 = - sin^2 x

36. (1/1+cos x) + (1/1-cos x )

1/(1+cos x) + 1/ (1 - cos x ) =
*sama kan penyebut

= 1(1-cos x ) + 1 (1+ cos x) / (1 +cos x )(1- cos x)
= ( 1 - cosx + 1 + cos x) / ( 1 - cos² x)
= (2) /(sin² x)
= 2 cosec² x 

37. bentuk sederhana dari 1-cos² x/cos² x + (1+tan² x) cos² x adalah​

Jawaban:

[1 - cos²x / cos²x] + (1 + tan²x) cos²x

= [sin²x / cos²x] + (sec²x) cos²x

= [tan²x] + (1/cos²x) cos²x

= tan²x + 1

= sec²x

semoga membantu

38. Tentukan perubahan cos 2X-cos x/cos x-1=2cos x+1

(cos 2x - cos x)/(cos x -1) = (2 cos²x - 1 - cos x)/(cos x - 1)
                                         = (2 cos x + 1)(cos x - 1)/(cos x - 1)
                                         = 2 cos x + 1

39. 1 / 1 + cos x + 1/ 1 - cos x

1/1 + cos x + 1/1 - cos x =
1/1 + 1/1 + cos x - cos x = 2


semoga terbantu :)

40. buktikan sin x*cos x*tan x=(1-cos x)(1+cos x)

sin²x + cos²x = 1
maka sin²x = 1-cos²x

sin x*cos x*tan x = sin x* cos x* sin x/cos x
                             = sin x * sin x
                              = sin² x
                             = 1- cos²x
                             = (1+ cos x ) (1-cos x) ==> terbukti



semoga membantu :)
Sin x . Cos x . Tan x = (1-Cos x) (1+Cos x)
Sin x . Cos c . Sin x / Cos x = 1-Cos ^2 x
Sin x . Sin x = 1-Cos^2 x
Sin^2 x= 1-Cos^2 x
Sin^2 x + Cos ^2 x = 1
1 = 1