solve 2+COS2X=3SIN2X ? show all steps please?

Answer 1

Could it be that question is:-
2 + cos ² x = 3 sin ² x
2 + 1 - sin ² x = 3 sin ² x
4 sin ² x = 3
sin ² x = 3 / 4
sin x = ±√3 / 2
x = 60° , 120° , 240° , 300°

Answer 2

Let y1 = 2 + cos2x
Let y2 = 3sin 2x
Plug thes equations into your calculator and note where the graphs intersect between 0 and 2pi. the points of intersection are apprximately 0.53,1.4, 3.67, and 4.54 radians

Answer 3

1) 2+cos(2x)=3sin(2x),

[2+cos(2x)]^2=[3sin(2x)]^2 ,

4+ 4·cos(2x)+ cos^2(2x) = 9 sin^2(x)

4+ 4·cos(2x)+ cos^2(2x) = 9 [1-cos^2(x)]

4+ 4·cos(2x)+ cos^2(2x) = 9 [1-cos^2(x)]

10· cos^2(2x) + 4·cos(2x) - 5 = 0

cos(2x) = [-2 ±√(4+50)]/10 =[-2 ±3√6]/10

*cos(2x) = [-2 +3√6]/10
--> x=28º50' +180º·k or
--> ¡ the other isn't solution x= 151º10' + 180º·k!

*cos(2x)=[-2-3√6]/10
--> x=79º36'6'' +180º·k

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

2) 2+cos^2x=3sin^2x

2+ 1 -sin^2x =3·sin^2x

4·sin^2x=3
sin^2x=3/4

sinx = ±√3 / 2 -->

x=60º+ 360º · k
x=120º+360º·k
x=240º+360º·k
x=300+360º·k with k any integer.

Saludos.

Answer 4

y'= 4xln(x-1)+(2x2-7)/(x-1)
y''= 4ln(x-1)+ 4x/(x-1) + 4x(x-1)-2x2+7 = 4ln(x-1) + 6x2-8x+7
(x-1)2 (x-1)2
y'''= 4/(x-1) + (12x-8)(x-1)2-2(x-1)( 6x2-8x+7) = 4x2-12x-2
(x-1)4 (x-1)3
y''''= (8x-12)(x-1)3-3(x-1)2(4x2-12x-2) = -4x2+16x+18
(x-1)6 (x-1)4
y'''''= (-8x+16)(x-1)4-4(x-1)3(-4x2+16x+18) = 8x2-40x-88
(x-1)8 (x-1)5
18.2.
y'= -2xln2x + 2lnx(3-x2)
x
y''= -2ln2x-4xlnx+-4x2lnx+2(3-x2)-2(3-x2)lnx =
x x2
= -2ln2x–4lnx - 2x2lnx+6lnx+2x2-6
x2
y'''= -4lnx – 4/x – (4xlnx+2x+6/x+4x)x2-2x(2x2lnx+6lnx+2x2-6) =
x x4
= 12lnx-4x2lnx-6x2-18
x3
18.3.
y'= cosx2-2x2sinx2
y''= -2xsinx2-4xsinx2-4x3cosx2 = -6xsinx2 – 4x3cosx2
y'''= -6sinx2-12x2cosx2-12x2cosx2+8x4sinx2 = 8x4sinx2-6sinx2-24x2cosx2
18.4.
y'= √(x-1)/(x-1) – ln(x-1)/2√(x-1) = 2-ln(x-1)
x-1 2(x-1)3/2
y''= -2√(x-1)-3√(x-1)(2-ln(x-1)) = 3√(x-1)ln(x-1)-8√(x-1)
4(x-1)3 4(x-1)3
y'''=((3ln(x-1))/(2√(x-1))+3√(x-1)/(x-1)-8/2√(x-1))(x-1)3-3(x-1)2(3√(x-1)ln(x-1)-8√(x-1)) =
4(x-1)6
= 46√(x-1)-15√(x-1)ln(x-1)
8(x-1)4
18.5.
y'= x2/lnx-3x2log2x = 1-3ln2log2x
x6 ln2 x4 ln2
y''= -3x3-4x3(1-3ln2log2x) = 12ln2log2x-7
x8 ln2 x5 ln2
y'''= 12x4+5x4(12ln2log2x-7) = 60ln2log2x-23
x10 ln2 x6 ln2
18.6.
y'= 12x2e2x+1+2(4x3+5)e2x+1= (8x3+12x2+10)e2x+1
y''= (24x2+24x) e2x+1+2(8x3+12x2+10)e2x+1=(16x3+48x2+24x+20) e2x+1
y'''= (48x2+96x+24)e2x+1+2(16x3+48x2+24x+20)e2x+1= 16(2x3+9x2+9x+4)e2x+1
y''''= 16((6x2+18x+9)e2x+1+2(2x3+9x2+9x+4)e2x+1)= 16(6x3+24x2+36x+17)e2x+1
y'''''= 16((18x2+48x+36)e2x+1+2(6x3+24x2+36x+17)e2x+1)= 16(12x3+72x2+120x+70)e2x+1
18.7.
y'= 2xsin(5x-3)+5x2cos(5x-3)
y''= 2sin(5x-3)+10xcos(5x-3)+10xcos(5x-3)-25x2sin(5x-3) = 2sin(5x-3)+20xcos(5x-3)-
-25x2sin(5x-3)
y'''= 10cos(5x-3)+20cos(5x-3)-100xsin(5x-3)-50xsin(5x-3)-125x2sin(5x-3)= 30cos(5x-3)-
-150xsin(5x-3)-125x2sin(5x-3)
18.8.
y'= x-2xlnx = 1-2lnx
x4 x3
y''= -2x2-3x2(1-2lnx) = -5+6lnx
x6 x4
y'''= 6x3-4x3(6lnx-5) = 26-24lnx
x8 x5
y''''= -24x4-5x4(26-24lnx) = 120lnx-154
x10 x6
18.9.
y'= 2ln2x+2lnx(2x+3)
x
y''= 4lnx/x+ 2(2x+3)+4xlnx-4xlnx-6lnx = 4xlnx+4x+6-6lnx
x2 x2
y'''= (4lnx+8-6/x)x2-2x(4xlnx+4x+6-6lnx) = 12lnx-4xlnx-18
x4 x3
18.10.
y'= 2xarctgx+1
y''= 2arctgx+2x/(1+x2)
y'''= 2/(1+x2)+ 2(1+x2)-4x2 = 4/(1+x2)2
(1+x2)2
18.11.
y'= x2-3x2lnx = 1-3lnx
x6 x4

y''= -3x3-4x3(1-3lnx) = -7+12lnx
x8 x5
y'''= 12x4-5x4(12lnx-7) = -23-60lnx
x10 x6
y''''= -60x5+6x5(60lnx+23) = 360lnx+78
x12 x7
18.12.
y'= 4*2-x-(4x+3)2-xlnx
y''= -4ln2*2-x-4ln2*2-x+(4x+1)ln22*2-x= 2-xln2(4xln2+ln2-8)
y'''= -2-xln22(4xln2+ln2-8)+4*2-xln22= 2-xln22(12-4xln2-ln2)
y''''= -2-xln32(12-4xln2-ln2)-4*2-xln32= 2-xln32(4xln2+ln2-16)
y'''= -2-xln42(4xln2+ln2-16)+4*2-xln42= 2-xln42(ln2-4xln2-12)
18.13.
y'= -2e1-2xsin(2+3x)+3e1-2xcos(2+3x)= e1-2x(3cos(2+3x)-2sin(2+3x))
y''= -2e1-2x(3cos(2+3x)-2sin(2+3x))+e1-2x(-9sin(2+3x)-6cos(2+3x))= e1-2x(-12cos(2+3x)-5sin(2+3x))
y'''= -2e1-2x(-12cos(2+3x)-5sin(2+3x))-e1-2x(-36sin(2+3x)+15cos(2+3x))= e1-2x(46cos(2+3x)+9sin(2+3x))
y''''= -2e1-2x(46cos(2+3x)+9sin(2+3x))+e1-2x(-27sin(2+3x)+138cos(2+3x))= e1-2x(120cos(2+3x)-119sin(2+3x))
18.14.
y'= 1-ln(3+x)
(3+x)2
y''= -(3+x)-2(3+x)(1-ln(3+x)) = -3+2ln(3+x)
(3+x)4 (3+x)3
y'''= 2(3+x)2-3(3+x)2(2ln(3+x)-3) = 11-6ln(3+x)
(3+x)6 (3+x)4
18.15.
y'= 6x2cosx-2x3sinx-sinx
y''= 12xcosx-12x2sinx-2x3cosx-cosx
y'''= 12cosx-36xsinx-18x2cosx+2x3sinx+sinx
y''''= 2x3cosx+18x2sinx+6xsinx+cosx-48sinx-72xcosx
y'''''= 24x2cosx-2x3sinx+108xsinx+6xcosx+5sinx-120cosx
18.16.
y'= 2xln(x-3)+x2+3
x-3
y''= 2ln(x-3) + 2x/(x-3) + 2x2-6x-x2-3 = 2ln(x-3) + 3x2-12x-3
(x-3)2 (x-3)2
y'''= 2/(x-3) + (6x-12)(x-3)-2(x-3)( 3x2-12x-3) = -4x2+18x+12
(x-3)4 (x-3)3
y''''= (-8x+18)(x-3)3-2(x-3)2(-4x2+18x+12) = 6x-78
(x-3)6 (x-3)4
18.17.
y'= 1/2*e(x-1)/2(1-x-x2)+ e(x-1)/2(-1-2x)= 1/2* e(x-1)/2(-1-5x-x2)
y''= 1/4*e(x-1)/2(-1-5x-x2)+ 1/2*e(x-1)/2(-5-2x)= 1/4* e(x-1)/2(-11-9x-x2)
y'''= 1/8*e(x-1)/2(-11-9x-x2)+ 1/4*e(x-1)/2(-9-2x)= 1/8* e(x-1)/2(-29-13x-x2)
y''''= 1/16*e(x-1)/2(-29-13x-x2)+ 1/8*e(x-1)/2(-13-2x)= 1/16* e(x-1)/2(-55-17x-x2)
18.18.
y'= 2xcos2x+sin2x
x2
y''= (2cos2x-4xsin2x+cos2x)x2-2x(2xcos2x+sin2x) = -xcos2x-4x2sin2x-2sin2x
x4 x3
y'''= (-cos2x+2xsin2x-8xsin2x-8x2cos2x-4cos2x)x3-3x2(-xcos2x-4x2sin2x-2sin2x) =
x6
= 6x2sin2x-8x2cos2x-2xcos2x+6sin2x
x4
18.19.
y'= ln(x+4) +(x+7)/(x+4)
y''= x+4+x+4-x-7 = x+1
(x+4)2 (x+4)2
y'''= (x+4)2-2(x+1)(x+4) = 2-x
(x+4)4 (x+4)3
y''''= -(x+4)3-3(x+4)2(2-x) = 2x-10
(x+4)6 (x+4)4
y'''''= 2(x+4)4-4(x+4)3(2x-10) = 48-6x
(x+4)8 (x+4)5
18.20.
y'= 3*3-x-(3x-7)3-xln3= 3-x(3-3xln3+7ln3)
y''= -3-xln3(3-3xln3+7ln3)+3*3-xln3= 3-x ln23(7-3x)
y'''= -3-xln33(7-3x)-3*3-xln23= 3-x ln23(3xln3-7ln3-3)
y'''= -3-xln33(3xln3-7ln3-3)+3*3-xln33= 3-x ln33(3xln3-7ln3+6)
18.21.
y'= 1-2ln(2x+5)
(2x+5)2
y''= -2(2x+5)-2(2x+5)( 1-2ln(2x+5)) = -4+4ln(2x+5)
(2x+5)4 (2x+5)3
y'''= 4(2x+5)2-3(2x+5)2(4ln(2x+5)-4) = -8-12ln(2x+5)
(2x+5)6 (2x+5)4
18.22.
y'= 1/2*ex/2sin2x+2ex/2cos2x= ex/2/2*(sin2x+4cos2x)
y''= ex/2/4*(sin2x+4cos2x)+ ex/2/2(2cos2x-8sin2x)= ex/2/4*(-15sin2x+8cos2x)
y'''= ex/2/8*(-15sin2x+8cos2x)+ ex/2/4(-16cos2x-30sin2x)= ex/2/8*(-45sin2x-24cos2x)
y''''= ex/2/16*(-45sin2x-24cos2x)+ ex/2/8(-90cos2x+48sin2x)= ex/2/16*(51sin2x-204cos2x)
18.23.
y'= x4-5x4lnx = 1-5lnx
x5 x
y''= -5-1+5lnx = 5lnx-6
x2 x2
y'''= 5x-2x(5lnx-6) = 17-10lnx
x4 x3
18.24.
y'= ln(1-3x)-3x/(1-3x)
y''= 1/(1-3x) – 3(1-3x)+9x = -3x-2
(1-3x)2 (1-3x)2
y'''= -3(1-3x)2+2(1-3x)(3x+2) = 15x+1
(1-3x)4 (1-3x)3
y''''= 15(1-3x)3-3(1-3x)2(15x+1) = 12-90x
(1-3x)6 (1-3x)4
18.25.
y'= 3e3x+2(x2+3x+1)+e3x+2(2x+3)= e3x+2(3x2+11x+6)
y''= 3e3x+2(3x2+11x+6)+e3x+2(6x+11)= e3x+2(9x2+39x+29)
y'''= 3e3x+2(9x2+39x+29)+e3x+2(18x+39)= e3x+2(27x2+135x+126)
y''''= 3e3x+2(27x2+135x+126)+e3x+2(54x+135)= e3x+2(81x2+459x+513)
y'''''= 3e3x+2(81x2+459x+513)+e3x+2(162x+459)= e3x+2(243x2+1539x+1998)
18.26.
y'= -2-xln2(5x-8)+5*2-x= 2-x(5-5xln2+8ln2)
y''= -2-xln2(5-5xln2+8ln2)-5*2-xln2= 2-xln2(8ln2-10-5xln2)
y'''= -2-xln2(5-5xln2+8ln2)-5*2-xln22= 2-xln2(-13ln2+10+5xln2)
y''''= -2-xln22(-13ln2+10+5xln2)+5*2-x ln22= 2-x ln22(13ln2-5-5xln2)
18.27.
y'= 1-ln(x-2)
(x-2)2
y''= -(x-2)2-2(x-2)(1-ln(x-2)) = -x+2ln(x-2)
(x-2)4 (x-2)3
y'''= 2(x-2)2-(x-2)3-3(x-2)2(-x+2ln(x-2)) = 2x+4-6ln(x-2)
(x-2)6 (x-2)4
y''''= 2(x-2)4-6(x-2)3-4(x-2)3(2x+4-6ln(x-2)) = 24ln(x-2)-6x+14
(x-2)8 (x-2)5
y'''''= 24(x-2)4-6(x-2)5-5(x-2)4(24ln(x-2)-6x+14) = 24x-34-120ln(x-2)
(x-2)10 (x-2)6

18.28.
y'= -e-x(cos2x-3sin2x)+e-x(-2sin2x-6cos2x)= e-x(sin2x-7cos2x)
y''= -e-x(sin2x-7cos2x)+e-x(14sin2x+2cos2x)= e-x(13sin2x+9cos2x)
y'''= -e-x(13sin2x+9cos2x)+e-x(-18sin2x+26cos2x)= e-x(-31sin2x+15cos2x)
y''''= -e-x(-31sin2x+15cos2x)+e-x(-30sin2x-62cos2x)= e-x(sin2x-77cos2x)
18.29.
y'= 5ln2x+2lnx(5x-1)
x
y''= 10lnx/x+2(5x-1)+2xlnx(5x-1) = 10x2lnx+8xlnx+10x-2
x2 x2
y'''= 20x3lnx+10x3+8x2lnx+8x2+10x2 = 20xlnx+10x+8lnx+18
x4 x2
18.30.
y'= 1-2ln3log3x
x3ln3
y'' = -2x2-3x2(1-2ln3log3x) = -5-6ln3log3x
x6ln3 x4ln3
y''' = -6x3+4x3(5+6ln3log3x) = 14+24ln3log3x
x8ln3 x5ln3
y'''' = 24x4-5x4(14+24ln3log3x) = -46-120ln3log3x
x10ln3 x6ln3
18.31.
y'= 3x2e4x+3+4e4x+3(x3+3)= e4x+3(4x3+3x2+12)
y''= 4e4x+3(4x3+3x2+12)+e4x+3(12x2+6x)= e4x+3(16x3+24x2+6x+12)
y'''= 4e4x+3(16x3+24x2+6x+12)+e4x+3(48x2+48x+6)= e4x+3(64x3+144x2+72x+54)
y''''= 4e4x+3(64x3+144x2+72x+54)+e4x+3(192x2+288x+72)= e4x+3(256x3+768x2+576x+288)

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