Sinx/cos^2x/2=4sin^2x/2

Sinx/cos^2x/2=4sin^2x/2

∫4sin x cos^2 2x dx =....

1. ∫4sin x cos^2 2x dx =....


[tex] \frac{\int\limits {4 sinx} \, cos ^{2} 2x}{sinx} [/tex]
[tex] \int\limits{4 cos^{2} 2x } \, dx [/tex]
[tex] \frac{4}{2} sin ^{2} 2x [/tex]
[tex]2 sin^{2} 2x[/tex]
sepertinya begini, maaf kalau salah

2. Cos 2x + 5 sinx +2 =0


cos 2x = 1 - 2sin^2x

cos 2x + 5sinx + 2 =0
1 - 2sin^2x + 5sinx + 2 =0
-2 sin^2x + 5sinx +3

misal sin x = a
- 2a^2 + 5a + 3 =0
faktorkan:
(-a +3) (2a + 1)

sin x = 3 atau sin x = -1/2

sinx = 3 tidak mungkin karena nilai sin maksimal adalah 1

maka sin × = -1/2
× = 210 atau 330 derajat

3. (sinx+cosx)(sinx-cosx)=a. 2 sin^2x-1b. 2 cos^2x-1c. 1-2sin^2xd. 1-2cos^2xe. 1+cos^2x


= (sinx+cosx)(sinx-cosx)
= sin²x - cos²x

#kemungkinan pertama

= sin²x - cos²x
= 1-cos²x - cos²x
= 1-2cos²x

#kemungkinan kedua

= sin²x - cos²x
= sin²x - (1-sin²x)
= 2sin²x -1

jawaban → A atau D= (sinx+cosx)(sinx-cosx)
= sin²x - cos²x

#kemungkinan pertama

= sin²x - cos²x
= 1-cos²x - cos²x
= 1-2cos²x

#kemungkinan kedua

= sin²x - cos²x
= sin²x - (1-sin²x)
= 2sin²x -1

jawaban → A

4. jika x memenuhi 3sinxcosx-sinx-6cosx+2=0 maka cos 2x=


misalkan :
sin x = a
cos x = b

3ab - a - 6b + 2 = 0
faktorkan
a + 6b - 3ab - 2 = 0
(a-2 ) (1-3b ) = 0
a = 2
sin x = 2 , salah , karena -1<sin x < 1
1 - 3b = 0
3b = 1
  b = 1/3
cos x = 1/3 , benar , karena -1 < cos x < 1

sin x = √(1 - cos²x)
         = √(1 - (1/3)²)
         = √(8/9)
         = 2/3 √2

cos 2x = cos²x - sin²x
           = (1/3)² - (2/3 √2)²
           = 1/9 - 4/9 (2)
           = 1/9 - 8/9
           = -7/9
[tex]\displaystyle \text{misal :}\\\sin x=a\\\cos x=b\\\\3\sin x\cos x-\sin x-6\cos x+2=0\\3ab-a-6b+2=0\\a(3b-1)-2(3b-1)=0\\(a-2)(3b-1)=0\\a=2(\text{tidak mungkin})\text{ atau }b=\frac13\\\\\cos x=\frac13\\\\\cos2x=2\cos^2x-1\\\cos2x=2\left(\frac13\right)^2-1\\\cos2x=\frac29-\frac99\\\boxed{\boxed{\cos2x=-\frac79}}[/tex]

5. Turunan dari f(x)= 1/2 sinx - 3 cos 2x


f(x) = ½sinx - 3cos2x
f'(x) = ½cosx - (-3sin2x × 2)
f'(x) = ½cosx + 6sin2x
f'(x) = ½cosx + 6 × 2sinxcosx
f'(x) = ½cosx + 12sinxcosx
f'(x) = cosx(½ + 12sinx)

6. buktikan sinx cosecx-sin^2x=cos^2x​


[tex] \sin(x) \csc(x) - { \sin(x) }^{2} = { \cos(x) }^{2} [/tex]

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

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

[tex] { \cos(x) }^{2} = { \cos(x) }^{2} [/tex]

semoga membantu^^


7. sinx - cos^2x . sinx / sin^2x ??? Please need untuk besok


( sin x - cos^2 x. Sin x) / sin^2 x
= [ sin x ( 1 - cos^2 x)] / sin^2 x
= (1 - cos^2 x) / sin x
= sin^2 x / sin x
= sin x

8. seharusnya cos 2x= cos (x+x) dimana didapatkan cos 2x = cos x cos x - sinx sinx


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

9. Nilai cos x yang memenuhi persamaan 3cos^2 2x + 4sin (phi/2 - 2x) - 4 = 0 adalah......... sama caranya yaa. Makasiiii


3 cos²2x + 4 sin(π/2 - 2x) - 4 = 0
3 cos²2x + 4 sin(90° - 2x) - 4 = 0
3 cos²2x + 4 cos 2x - 4 = 0
(3 cos 2x - 2) (cos 2x + 2) = 0
cos 2x = 2/3 [memenuhi]
cos 2x = -2 [tidak memenuhi]

cos 2x = 2/3
1 - 2 sin²x = 2/3
1 - 2/3 = 2 sin²x
1/3 = 2 sin²x
1/3 . 1/2 = sin²x
sin²x = 1/6
sin²x + cos²x = 1
1/6 + cos²x = 1
cos²x = 1 - 1/6
cos²x = 5/6
cos x = √(5/6)
cos x = 1/6 √30

maka nilai cos x =  1/6 √30

10. sin²2x-2 sinx cos x-2=0


sin²2x-2 sinx cos x-2=0
sin²2x- sin2x-2=0
(sin2x - 2) (sin2x +1)
sin2x = 2
(tak ada sin bernilai 2)
sin2x = -1
2x = 270°
x = 135°

11. Jika f(x) = (cos x + sin x) ^2 maka f'(x) adalah... a. 2 cos 2x b. 2 sin 2x c. cos 2x d. sin 2x e. cos 2x- sinx jawab dengan penjelasan yahh


[tex]f(x)=(cosx+sinx)^2 \\ f(x)=cos^2x+2cosxsinx+sin^2x \\ f(x)=1+2cosxsinx \\ f(x)=1+sin2x \\ \\ f'(x)=1\times0x^{0-1}+2\times cos2x \\ f'(x)=0+2cos2x \\ f'(x)=2cos2x \\ \\ \text{Maaf kalau salah}[/tex]

12. cosx-sinx=1/2 tentukan cos 2x


cos x - sin x = 1/2 --> kuadratkan
(cos x - sin x)² = (1/2)²
cos² x + sin²x - 2 sinx cos x = 1/4
(cos² x+sin² x) - sin 2x = 1/4
1 - sin 2x = 1/4
sin 2x = 1 - 1/4 = 3/4

cos 2x = √(1 -sin² 2x) = √(1 -(3/4)²)
cos 2x = √(1-9/16) = √(7/16) 
cos 2x = 1/4 √7  atau cos 2x = - 1/4√7


13. nilai dari (4sin 2x-cos x)dx adalah.....


Pembahasan:

∫(4 sin 2x - cos x) dx

= -4/2 cos 2x - sin x + C

= -2 cos 2x - sin x + C

14. Sederhanakan 1 - 4sin 2x . Cos 2x = 1 - 2sin x . Cos x


1 - 4sin 2x.cos 2x = 1 - 2sin x.cos x
-4sin 2x.cos 2x = -2sin x.cos x
4sin 2x.cos 2x = sin 2x
4cos 2x = 1
cos 2x = ¼

15. sinx . cosecx - sin^2X = cos^2x


Sinx. Cosec x - Sin^2x

= Sinx (1/sinx) - (1- cos^2x)

= 1 - (1 -Cos^2x)

= 0+ Cos^2x

= cos^2x (Terbukti)

16. hasil dari integral 4sin 4x cos 2x dx


[tex]\int \:4\sin \left(4x\right)\cos \left(2x\right)dx=2\left(-\frac{\cos \left(2x\right)}{2}-\frac{\cos \left(6x\right)}{6}\right)+C[/tex]

17. Nilai cos x yang memenuhi persamaan 3cos^2 2x + 4sin (phi/2 - 2x) - 4 = 0 adalah......... sama caranya yaa. Makasiiii


3 cos²2x + 4 sin(π/2 - 2x) - 4 = 0
3 cos²2x + 4 sin(90° - 2x) - 4 = 0
3 cos²2x + 4 cos 2x - 4 = 0
(3 cos 2x - 2) (cos 2x + 2) = 0
cos 2x = 2/3 [memenuhi]
cos 2x = -2 [tidak memenuhi]

cos 2x = 2/3
1 - 2 sin²x = 2/3
1 - 2/3 = 2 sin²x
1/3 = 2 sin²x
1/3 . 1/2 = sin²x
sin²x = 1/6
sin²x + cos²x = 1
1/6 + cos²x = 1
cos²x = 1 - 1/6
cos²x = 5/6
cos x = √(5/6)
cos x = 1/6 √30

maka nilai cos x =  1/6 √30

18. lim (2 sinx . cos 2x)x→π/4 ​


Penjelasan dengan langkah-langkah:

_____________________________

LIMIT FUNGSI TRIGONOMETRI

[tex]\sf \lim \limits_{x \to \frac{\pi}{4}} 2\sin x \cos 2x[/tex]

[tex]= \sf 2 \sin \left(\dfrac{\pi}{4}\right)\cos \left(2 \times \dfrac{\pi}{4}\right)[/tex]

[tex]= \sf 2 \sin \left(\dfrac{\pi}{4}\right) \cos \left( \dfrac{\pi}{2}\right)[/tex]

[tex]= \sf 2 \times \dfrac{1}{2}\sqrt{2} \times 0[/tex]

[tex]= \sf 0[/tex]


19. cos^2x - son^2x /cosx -sinx


jadi 2cosx - 2sinx / cosx- sinx jadi jawabannya 2

20. 1)  nilai x dari sin(2x-30)=cos 20 2) NIlai x dari cos 2x =sinx


mungkin seperti ini... :)
sin(2x - 30) = cos 20
sin(2x - 30) = sin(90 - 20)
2x - 30 = 90 - 20 + k.360
2x - 10 = 90 + k.360
2x = 100 + k.360
x = 50 + k.180
k = 0 ===> x = 50

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