buktikan bahwa : tan^6x = tan^4x . sec^2x - tan^2x . sec^2x + sec^2 - 1
1. buktikan bahwa : tan^6x = tan^4x . sec^2x - tan^2x . sec^2x + sec^2 - 1
Penjelasan dengan langkah-langkah:
Sifat trigonometri yang digunakan:
[tex]sec^2(x) = tan^2(x) +1[/tex] atau [tex]sec^2(x) - 1= tan^2(x)[/tex]
[tex]tan^6(x) = (tan^2(x))^3\\\\tan^6(x) = (sec^2(x)-1)^3\\\\tan^6(x) = (sec^2(x))^3-3(sec^2(x))^2+3sec^2(x)-1\\\\tan^6(x) = (sec^2(x))^3-3(sec^2(x))^2+2sec^2(x)+sec^2(x)-1\\\\tan^6(x) = [sec^2(x)]\:[(sec^2(x))^2-3sec^2(x)+2] + sec^2(x)-1\\\\tan^6(x) = [sec^2(x)]\:[(sec^2(x)-2)(sec^2(x)-1)]+sec^2(x)-1\\\\tan^6(x) = [sec^2(x)]\:[(sec^2(x)-1)(tan^2(x))]+sec^2(x)-1\\\\tan^6(x) = [sec^2(x)]\:[(sec^2(x)-1-1)(tan^2(x))]+sec^2(x)-1\\\\tan^6(x) = [sec^2(x)]\:[(tan^2(x)-1)(tan^2(x))]+sec^2(x)-1\\\\[/tex]
[tex]tan^6(x) = [sec^2(x)]\:[tan^4(x)-tan^2(x)]+sec^2(x)-1\\\\tan^6(x) =tan^4(x)sec^2(x)-tan^2(x)sec^2(x)+sec^2(x)-1[/tex]
(Terbukti)
Maka, terbukti bahwa [tex]tan^6(x) =tan^4(x)sec^2(x)-tan^2(x)sec^2(x)+sec^2(x)-1[/tex]
2. [tex]5.\ Bentuk\ Sederhana\ Dari\ \frac{1\ +\ tan\ x}{1\ -\ tan\ x}[/tex] Objektifnya : A. sec 2x - tan x B. sec x - tan 2x C. sec x + tan x D. sec x + tan 2x E. sec 2x + tan 2x
[tex]\huge \frac{1+\tan~x}{1-\tan~x}[/tex]
[tex]=\frac{1+\frac{\sin~x}{\cos~x}}{1-\frac{\sin~x}{\cos~x}}[/tex]
[tex]=\frac{\frac{\cos~x+\sin~x}{\cos~x}}{\frac{\cos~x-\sin~x}{\cos~x}}[/tex]
[tex]=\frac{\cos~x+\sin~x}{\cos~x-\sin~x}[/tex]
[tex]=\frac{\cos~x+\sin~x}{\cos~x-\sin~x}\times \frac{\cos~x+\sin~x}{\cos~x+\sin~x}[/tex]
[tex]=\frac{\cos^2~x+2.\cos~x.\sin~x+\sin^2~x}{\cos^2~x-\sin^2~x}[/tex]
[tex]=\frac{1+\sin~2x}{\cos~2x}[/tex]
[tex]=\frac{1}{\cos~2x}+\frac{\sin~2x}{\cos~2x}[/tex]
[tex]\huge =\sec~2x+\tan~2x[/tex]
[tex]\huge \to \sf (~E~)[/tex]
3. Bentuk (sin^4x-cos^4x)/(sec^4x-tan^4x) ekuivalen dengan…… a. (cos^2x+sin^2x)/(sec^2x-tan^2x) b. (cos^2x-sin^2x)/(sec^2x-tan^2x) c. (sin^2x-cos^2x)/(sec^2x-tan^2x) d. (sin^2x+cos^2x)/(sec^2x+tan^2x) d. (sin^2x+cos^2x)/(sec^2x+tan^2x)
Jawab:
Jawabannya ada di foto ya, semoga bermanfaat yaa
Penjelasan dengan langkah-langkah:
Sudah di jelaskan ya. Selamat belajar
4. sec^4 x - tan^4 x = sec^2x+ tan^2x
inget!!
kalo :
sec^4x kan = 1/cos^4 x
tan^4x = sin^4x / cos^4x
sin²x + cos²x = 1
berarti
sec^4x - tan^4x = sec²x + tan²x
(1/cos^4x) - (sin^4x/cos^4x) = sec²x + tan²x
(1-sin^4x)/cos^4x = sec²x + tan²x
(cos²x + sin²x - sin²x. sin2x) /(cos²x. cos²x) = sec²x + tan²x
(cos²x + sin²x(1-sin²x)) / (cos²x cos²x) = sec²x + tan²x
(cos²x + sin²x (cos²x)) / (cos²x cos²x) = sec²x + tan²x
(cos²x (1+sin²x)) / (cos²x cos²x) = sec²x + tan²x
(1+ sin²x) / cos²x = sec²x + tan²x
(1/cos²x) + (sin²x/cos²x) = sec²x + tan²x
sec²x +tan²x =sec²x +tan²x
terbukti tuh!!
5. integral tan⁴ 2x sec² 2x dx adalah....
Jawaban:
Ada pada Gambar
Penjelasan dengan langkah-langkah:
Ada pada Gambar
6. buktikanlah identitas sec^2x (1 - cos^2x) =tan^2x
Penjelasan dengan langkah-langkah:
sec²x(1-cos²x)
sec²x(sin²x)
1/cos²x (sin²x)
sin²x/cos²x
tan²x
terbukti
7. sin^2x + cos^2x/ sec^2x - tan^2x = 1
= (sin² x + cos² x)/(sec² x - tan² x)
= (sin² x + cos² x) / {1/ cos² x - sin² x/ cos² x}
= (sin² x + cos² x) / {( 1- sin² x)/(cos² x)}
= (sin² x + cos² x) / {(cos² x/ cos² x)}
= 1/ 1
= 1
..
8. integral x sec² (1+2x²) tan (1+2x²) dx
Jawab:
set
u = 1 + 2x²
du = 4x dx
x dx = 1/4 du
∫x sec² (1+2x²) tan (1+2x²) dx
= 1/4 ∫ tan u sec² u d u
set
v = tan u
dv = sec² u du
= 1/4 ∫ tan u sec² u d u|
= 1/4 ∫ v dv
= 1/4. 1/2 . v² + c
= 1/8 v² + c
= 1/8 tan² u + c
= 1/8 tan² (1 + 2x² ) + c
9. Integral (tan 2x + sec 2x)2
[tex]$\begin{align}&\int (\tan2x+\sec2x)^2\, dx\\&=\int\tan^2x+2\tan2x\sec2x+\sec^22x\, dx \\ &=\int (\sec^22x-1)+2\tan2x\sec2x+\sec^22x\, dx \\ &=\int2\sec^2(2x)\, dx+\int 2\tan(2x)\sec(2x)-1\, dx \\ &=\int \sec^2(2x)(2\,dx)+\int\tan(2x)\sec(2x)(2x\,dx)-\int dx \\ &=\tan(2x)+\sec(2x)-x+C\end{align}[/tex]
10. berapakah integral dari ∫(tan 2x + sec 2x)² dx
Mapel : Matematika
Materi : Integral
11. sec^4x–tan^4x=sec^2x+tan^2x buktikan identitas trigonometri
sec^4x - tan^4x = 1/cos^4x - sin^4x/cos^4x
= (1 - sin^4x)/cos^4x
= (cos^2x + sin^2x - sin^2x.sin^2x)/(cos^2x.cos^2x)
= (cos^2x + sin^2x(1 - sin^2x))/(cos^2x.cos^2x)
= (cos^2x + sin^2x.cos^2x)/(cos^2x.cos^2x)
= cos^2x(1 + sin^2x)/(cos^2x.cos^2x)
= (1 + sin^2x)/(cos^2x)
= sec^2x + tan^2x
Q.E.D.
12. Integral (tan 2x + sec 2x)2
[tex]\int (tan2x+sec2x)^{2}dx[/tex]
=> [tex]\int (tan^{2}2x+sec^{2}2x+2tan2xsec2x)dx[/tex]
Integralkan masing masing, maka:
=> [tex]\int (tan^{2}2x+sec^{2}2x+2tan2xsec2x)dx[/tex]
=> [tex](\frac{1}{2}tan2x-x)+tan2x+(-\frac{1}{2}sec2x)+C[/tex]
=> [tex]\frac{3}{2}tan2x-x-\frac{1}{2}sec2x+C[/tex]
13. integral tan 2x dikali sec 2x
u = 2x
du = 2 dx
dx = 1/2 du
∫ tan 2x sec 2x dx = 1/2 ∫ tan u sec u du = 1/2 (sec u) + c
= 1/2 sec 2x + c
14. Buktikan sec^2x-1/cosec^2x-1=tan^4
~Identitas Trigono
(sec²x - 1) / (csc²x - 1) = tan⁴x
tan²x / cot²x = tan⁴x
tan²x / (1 / tan²x) = tan⁴x
tan²x . tan²x = tan⁴x
tan⁴x = tan⁴x
- s e m a n g a t
15. sec^4x - sec^2x=tan^4x + tan^2x
^ artinya pangkat
sec^4 x - sec² x
= sec² x . (sec² x - 1)
= (tan² x + 1) . tan² x
= tan^4 x + tan² x
terbukti
16. integral dari (tan 2x + sec 2x)²dx adalah..
integral (tan 2x + sec 2x)^2
= integral tan^2 (2x) + sec^2 (2x) + 2 tan2x sec2x
= integral {sec^2 (2x) - 1 + sec^2 (2x)} + integral 2 tan2x sec2x
= integral {2sec^2 (2x) -1} + integral 2 tan2x sec2x
karena turunan tan2x = 2sec^2 (2x) dan turunan sec2x = 2 tan2x sec2x, maka:
=> integral {2sec^ (2x)} - integral 1 + integral 2 tan2x sec2x
= tan2x - x + sec2x
semoga membantu
17. Jika p + Tan 2x = 1 , maka sec x adalah ..
P + tan²x = 1
tan²x = 1 - p
1 + tan²x = sec²x
1 + 1 - p = sec²x
2 - p = sec²x
√(2 - p) = sec x
maka sec x = √(2 - p)
18. Buktikan bahwa 5 tan^2x + 4 = 5 sec^2x – 1
5 tan^2x + 4 = 5 (sec^2x – 1) + 4
= 5 sec^2x – 5 + 4
= 5 sec^2x – 1 (terbukti)
5 tan² 2x + 4
5( sec² 2x - 1) + 4
5 sec² 2x - 5 + 4
5 sec² 2x - 1
terbukti
19. buktikan bahwa tan^2x-sec^2x=-1
Penjelasan dengan langkah-langkah:
tan² x - sec² x
= sin²x/cos² x - 1/cos² x
= (sin² x - 1)/cos² x
= (sin² x - (sin² x + cos² x))/cos² x
= (sin² x - sin² x - cos² x)/cos² x
= - cos² x/cos² x
= -1
Terbukti
20. integral (tan 2x+ sec 2x)² dx
semoga membantu. silahkan dicek kembali
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