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

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

Jawabannya ada pada paling bawah pada foto.
tolong dikoreksi jikalau ada kesalahan :)

Terima kasih :)

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

(1 + sin x )² - (1 - sin x)²
(1 + sin x )(1+sinx) - (1 - sin x)(1-sinx)
(1+2sinx+sin²x) - (1-2sinx+sin²x)
4sinx
(terbukti)

3. sin x -1)( sin x + 1 )=​

(sin x - 1)(sin x + 1)

= sin² x + sin x - sin x - 1

= sin²x - 1

4. hasil dari (Sin x-1)(sin x+1)=

Rumus awal: Sin a sin b = 1/2 [cos (a-b) - cos (a+b)]
sin (x-1) sin (x+1) = 1/2 [cos (-2) - cos (2x)]. Sekian. Semoga puas dan jadi jawaban terbaik. :)

5. 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

±+++++((((

6. Cos x 1 - sin x _____ = _______ 1 - sin x Cos x

Jawaban:

KAU TAU TUSUK PISANG KE LUBANG DAK

ITUAM DIE T A S U

7. Identitas trigonometri Buktikan bahwa : a. (1 + sin x)² - (1 - sin x)² = 4 sin x b. 4 sin² x - 1 / 2 sin x + 1 = 2 sin x-1 mohon bantuan dari teman-teman brainly :)

a. (1 + sin x)² - (1 - sin x)² = 4 sin x
    (1 + sin x)(1 + sin x) - (1 - sin x)(1 - sin x) = 4 sin x
    1 + 2 sin x + sin² x - (1 - 2 sin x + sin²x)= 4 sin x
                   4 sin x = 4 sin x

b. 4 sin² x - 1 / 2 sin x + 1 = 2 sin x-1
   (2 sin - 1)(2 sin x + 1) /2 sin x + 1 = 2 sin x-1
    (2 sin - 1) = (2 sin - 1)
semoga membantu

8. Toling buktikan Sin x / 1 + cos x + 1 + cos x / sin x = 2 / sin x

penyeleaaian terlampir:

9. hitunglah dari(sin x-1) - (sin x+1)?

sinx-1-(sinx+1)=sinx-1-sinx-1=-2

10. Sin x / 1 + cos x + 1 + cos x / sin x = 2 / sin x

Semoga bermanfaat....

11. 1-cos x/sin x + sin x/1-cos x

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

[tex]=\left(\csc \left(x\right)-\cot \left(x\right)\right)+\frac{\sin \left(x\right)}{-\cos \left(x\right)+1}[/tex]

[tex]=-\cot \left(x\right)+\csc \left(x\right)+\frac{\sin \left(x\right)}{1-\cos \left(x\right)}[/tex]

[tex]=\frac{\sin \left(x\right)}{1-\cos \left(x\right)}-\frac{\cot \left(x\right)\left(1-\cos \left(x\right)\right)}{1-\cos \left(x\right)}+\frac{\csc \left(x\right)\left(1-\cos \left(x\right)\right)}{1-\cos \left(x\right)}[/tex]

[tex]=\frac{\sin \left(x\right)-\cot \left(x\right)\left(1-\cos \left(x\right)\right)+\csc \left(x\right)\left(1-\cos \left(x\right)\right)}{1-\cos \left(x\right)}[/tex]

[tex]=\frac{\sin \left(x\right)-\cot \left(x\right)\left(-\cos \left(x\right)+1\right)+\left(\csc \left(x\right)-\cot \left(x\right)\right)}{1-\cos \left(x\right)}[/tex]

[tex]=\frac{\sin \left(x\right)-\cot \left(x\right)\left(1-\cos \left(x\right)\right)-\cot \left(x\right)+\csc \left(x\right)}{1-\cos \left(x\right)}[/tex]

[tex]=\frac{\sin \left(x\right)-2\cot \left(x\right)+\cot \left(x\right)\cos \left(x\right)+\csc \left(x\right)}{1-\cos \left(x\right)}[/tex]

12. cos² x / 1 - sin x = 1 + sin x

Cos^2x =sin^2 x - 1
1 - sin x = - (sin x + 1)
= sin^2 x - 1 : 1 - sin ^x
= (sin x + 1) ( sin x - 1) : 1 - sin x
= (sin x + 1) -(1 - sin x) : 1 - sin x
= - sin x - 1 = sin x + 1@
cos² x /(1 - sin x) = 1 + sin x
(1 - sin² x) / (1 - sin x)   = 1 + sin x
(1 - sin x)(1 + sin x) / (1-sin x)  = 1 + sin x
1 + sin x = 1 + sinx

13. = Buktikanlah identitas trigonometri berikut. 1. (1 + sin x)(1 - sin x) = cos²x 2. (sin x - cos x)2 = 1 - 2 sin x cos X 3. 3 cos²x = 3-3 sin? 4. (sin x + cos x)(sin x + cos x) = 2 sina x-15. (sin x - cos x)2 + 2 cos x sin x = 1​

Jawaban:

[tex]1). \: \: (1 + sin \: x)(1 - sin \: x) = {cos}^{2} x \\buktikan \: ruas \: kiri \\ 1 - sin \: x + sin \: x - {sin}^{2} x \\ 1 - {sin}^{2} x \\ {cos}^{2} x[/tex]

[tex]2). \: \: {(sin \: x - cos \: x)}^{2} = 1 - 2sin \: x \: cos \: x \\ buktikan \: ruas \: kiri \\ {sin}^{2} x - 2sin \: x \: cos \: x + {cos}^{2} x \\ {sin}^{2} x \: + {cos}^{2} x - 2sin \: x \: cos \: x \\ 1 - 2sin \: x \: cos \: x[/tex]

[tex]3). \: \: 3 {cos}^{2} x = 3 - 3 {sin}^{2}x \\ buktikan \: ruas \: kiri \\ 3(1 - {sin}^{2} x) \\ 3 - 3 {sin}^{2} x[/tex]

[tex]4). \: (sin \: x + cos \: x)(sin \: x + cos \: x) = 2sin \: x \: - 1 \\ butikan \: ruas \: kiri \\ {sin}^{2} x \: + sin \: x \: cos \: x + sin \: x \: cos \: x + {cos}^{2} x \\ {sin}^{2} x + {cos}^{2} x + 2sin \: x \: cos \: x \\ 1 + 2sin \: x \: cos \: x \\ 1 + sin \: 2x[/tex]

14. Buktikan bahwa sin²x-1/tan x sin x - tan x = sin x +1/tan x

Penjelasan dengan langkah-langkah:

[tex] = \frac{ \sin {}^{2} (x) - 1 }{ \tan(x) \sin(x) - \tan(x) } [/tex]

[tex] = \frac{( \sin(x) + 1)( \sin(x) - 1) }{ \tan(x) ( \sin(x) - 1)} [/tex]

[tex] = \frac{( \sin(x) + 1) \cancel{( \sin(x) - 1)} }{ \tan(x) \cancel{( \sin(x) - 1)}} [/tex]

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

Terbukti.

15. (Sin x / 1-cos x ) + ( sin x / 1+cos x) adalah

Jawaban:

2 cosec x

Penjelasan dengan langkah-langkah:

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

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

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

= 2 sin x/sin² x

= 2 / sin x

= 2 . cosec x

Kode Kategorisasi : 10.2.6

Kelas 10

Pelajaran Matematika

Bab 6 - Trigonometri Dasar

16. (1 + sin x) ( 1 - sin x) = cos²x

[tex](1 + \sin \: x)(1 - \sin \: x) = \cos^{2} x \\ 1 - \sin^{2}x = \cos^{2} x \\ 1 = \cos^{2} x + \sin^{2}x \\ 1 = 1[/tex]
semoga membantu ^^

17. sin x/1-cos x=1+cos x/sin x

maaf kalau salah,,, skdar nyoba

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

Jawab:

Pembuktian terlampir di gambar.

19. Cos x per 1+ sin x + 1+ sin x per cos X

(cos x / 1+sin x) + (1+sin x / cos x)
jadi jika a/b + b/a = (a² + b²)/ba
(cos²x + 1 + 2sin x + sin²x) / cos x(1+sin x)
= (cos²x+sin²+1+2.sin x) / cos x(1+sin x)
= (1 + 1 + 2.sin x) / cos x(1 + sin x)
= (2 + 2.sin x) / cos x(1 + sin x)
= 2(1 + sin x) / cos x(1 + sin x)
= 2 / cos x
= 2 (1 / cos x)
= 2 sec x

20. 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

21. cos x / sin x + 1 + cos x / sin x - 1 ?

[tex] - 2 \tan(x) [/tex]

22. 1 + sin x° : 1 - sin x° = 3

Bab Trigonometri
Matematika SMA Kelas X

(1 + sin x)/(1 - sin x) = 3
1 + sin x = 3 (1 - sin x)
1 + sin x = 3 - 3 sin x
sin x + 3 sin x = 3 - 1
4 sin x = 2
sin x = 2/4
sin x = 1/2
x = 30°, 150°

HP = { 30°, 150° }

23. 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]

24. Sin²x/1-sin²x + cos²x/1-cos²x =

Jawab:

=[tex]\frac{tan^{4}x+1}{tan^{2}x}[/tex]

Penjelasan dengan langkah-langkah:

Catatan sebelum pengerjaan :

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

[tex]tan x =\frac{sin x}{cos x}[/tex]

[tex]\frac{1}{tan x} = cot x =\frac{cos x}{sin x}[/tex]

Cara Pengerjaan

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

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

=[tex]tan^{2}x}+\frac{1}{tan^{2}x}[/tex]

=[tex]\frac{tan^{4}x+1}{tan^{2}x}[/tex]

----------------------------------------------------------------------------------------------

Mapel :  Matematika

Kelas :  10

Materi :  Identitas Trigonometri

Kata kunci :  sin, cos, tan, cotangen, cot, sin^2, cos^2

Kode soal :  10.2

Kode Kategorisasi : 10.2.3

25. (1 + sin x / cos x) + (cos x / 1 + sin x)

Bentuk sederhana dari

(1+sin x)/ cos x  + (cos x)/ (1 +sin x) =
= {(1+sin x)(1+sin x)  + cos x (cos x)} / ( cos x )(1+ sin x)
= (1 + 2 sin x + sin² x + cos² x) / cos x(1+sin x)
= (1 + 2 sin x + 1 ) / cos x(1 + sin x)
= (2 + 2 sin x) / cos x(1+sin x)
= 2(1+sin x) / cos x (1+sin x)
= 2/cos x
= 2 sec x

(1 + sin x)/(cos x) + (cos x)/(1 + sin x) . (1 - sin x)/(1 - sin x)
= (1 + sin x)/(cos x) + (cos x(1 - sin x))/(1 - sin^2 x)
= (1 + sin x)/(cos x) + (cos x(1 - sin x))/(cos^2 x)
= (1 + sin x)/(cos x) + (1 - sin x)/(cos x)
= (1 + sin x + 1 - sin x) / (cos x)
= 2 / cos x
= 2 sec x

26. Cos x:1+sin x + 1+ sin x :cosx

cosx/(1+sin x) + (1+sin x)/cos x
= {cos x(cos x) + (1+sin x)(1+sin x)} / cos x(1+sin x)
= {cos²x + 1 + 2sin x + sin²x} / cos x(1 + sin x)
= {cos²x + sin²x + 1 + 2sin x} / cos x(1+sin x)
= {1 + 1 + 2sin x} / cos x(1 + sin x)
= (2 + 2sin x) / cos x(1 + sin x)
= 2(1 + sin x) / cos x(1 + sin x)
= 2 / cos x
= 2 sec x

27. 1 + sin x / 1 - sin x = (secc x + tan x)^

(1+sin x)/(1-sin x) x (1+sin x)/(1+sin x)
= 1 + 2sin x + sin² x / 1 - sin²x
= 1 + 2sin x + sin²x / cos²x
= (1/cos²x) + (2sin x/cos²x) + (sin²x / cos²x)
= sec²x + ( 2sin x/cos x)/cos x + (tan²x)
= sec² x + 2tan x sec x + tan²x
= (sec x + tan x)² (terbukti)

28. (sec x + tan x)² = 1+sin x : 1-sin x

jawab

(sec x + tan x)² = (1 + sin x) / (1 - sin x)

ruas kiri =
= (sec x + tan x)²
= (1/cos x   + sin x/cos x)²
= (1+ sin x) / cos x)²
= (1+ sin x)² / (cos² x)
= (1+ sin x )(1+sin x) / ( 1 - sin² x)
= (1+ sin x)(1+sin x) / (1 -sin x)(1 + sin x)
= (1+ sin x ) / (1+sin x) . (1+ sin x) /(1- sin x)
= (1+ sin x) / (1 -sin x)
= ruas kanan

29. sin x/1+cos x+Sin x /1-Cos x

Sin x/1+cos x+Sin x /1-Cos x = sin x/1 + sin x/1
                                               = 2 sin x/1
                                               = 2 sin x

30. (1 + sin x) (1 - Sin x) = 1/sec2x

note:

sin^2 + cos^2 x=1

secx=1/cosx

31. 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

32. 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

33. 1-cos x/sin x=sin x/1+cos x

-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-
[TRIGONOMETRI]
Kelas : 11
Kata kunci : Identitas Trigonometri 

Inti dari identitas trigonometri adalah :
sin²x+ cos²x  = 1
sin²x = 1-cos²x ... (1)
cos²x = 1-sin²x ... (2)

1-cosx  =  sin x   
 sin x      1+ cosx
Kalikan ruas kiri  dengan 1 + cosx .
1-cos x .(1+cosx) =     sin x   
sin x     (1+cosx)    (1+cos x)
⇒1 + cos x - cos x - cos²x =   sin x   
     sin x .(1+cosx)               (1+cosx)
⇒    1 - cos²x       =    sin x      ⇔kembali keidentitas diatas
    sin x (1+cosx)    (1+cos x)
⇒        sin²x          =   sin x      ⇔Coret sin x di ruas kiri
    sin x .(1+cos x)   (1+cos x)  
⇒       sin x   =     sin x      [TERBUKTI]
    (1+cos x)    (1+cos x)

-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-

34. (Sin x - 1) (sin x + 1)

(sin x - 1) (sin x + 1)
= sin²x + sin x - sin x - 1
= sin²x - 1

Ingat identitas ini :
sin²x + cos²x = 1

maka :
sin²x - 1 = -cos²x 

Jadi, (sin x - 1) (sin x + 1) = -cos²x

35. (sin x-cos x)(1 +sin x cos x)=sin³×cos³×

Jawaban:

(sin x - cos x)(1 + sin x cos x) = sin³ x - cos³ x

(sin x - cos x)(sin² x + cos² x + sin x con x) = sin³ x - cos³ x

sin³ x + sin x cos² x + sin² x cos x - sin² x cos x - cos³ x - sin x cos² x = sin³ x - cos³ x

sin³ x - cos³ x = sin³ x cos³ x

36. sin(n+1)x-sin(x-1)x=...

sin A - sin B = 2 cos 1/2 (A+B) sin 1/2 (A-B)

sin(x+1)x-sin(x-1)x

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

= 2cos 1/2(x²+x+x²-x) sin 1/2(x²+x-x²+x)

= 2cos 1/2(2x²) sin 1/2(2x)

= 2 cos x² sin x

37. Cos x/1-sin x=1 + sin x/cos x

saya belum belajar ini

38. (1+sin x) (1-sin x)= cos^x

jawab

(1 + sin x )(1- sin x )  = cos² x
1 - sin² x = cos x
cos x =  cos x
terbukti

39. Sin x/1+cos x + 1+cos x/sin x=

jadi jika ada a/b + c/d = (ad + bc)/bd

sin x / (1+cos x) + (1+cosx) / sin x
= (sin²x + 1 + 2.cos x + cos²x) / sin x(1+cos x)
= (sin²x+cos²x + 1 + 2cos x) / sin x(1+cos x)
= (1 + 1 + 2cos x) / sin x(1+cos x)
= (2 + 2 cos x) / sin x(1+cos x)
= 2(1 + cos x) / sin x(1 + cos x)
= 2 / sin x
= 2 (1/sin x)
= 2 cosec x

40. y = (1 + sin² x)(1 - sin²x)​

Jawaban:

y=1

Penjelasan dengan langkah-langkah:

y=(1+sini(×)2)-sin(×(×2)

y=1+sin(×)2-sin(×)2

y=1