Pre-calc midterm
Lne=x/x
0
e^0
1
lne
1
500e^-x=300
300/500 = 3/5 The flip of it because of e^-x 5/3
ln(x-1)=3.8
Convert 3.8 to a log (lne^a) ln(x-1)=lne^3.8 Ln's cancel out x-3=e^3.8 Get x by itself x=e^3.8-3
Log(base4)(3x)=3
Convert both to Logs Log(base4)(3x)=Log(base4)4^3 Log(base4) acts as 1 3x=64 x=64/3
lne^x^2
Convert from loga^xb to B logax x^2lne Lne=1 1x^2 x^2
3^(2x)=8
Convert to natural log 2xLn3=Ln80 Divide to get x by itself Ln80/2ln3
(Rad3)Secθ=2
Divide both sides by rad 3 Secθ=2/rad3 Sec identity to 1/cosθ=2/rad3 cosθ=rad3/2
e+1=0
FALSE
e^(2x)-4e^(x)-5
Factor 4 and 5 (e-5)(e+1) e-5=0 ln=5
3e^x+2=75
Get e by itself (by dividing out the 3) e^x+2=75 Convert to natural log (e = ln) lne^x+2=ln25 Get x by itself x=ln(25-2 ) Ans: Ln(25-2)
4^2x-7=64
Get even bases then solve for x with the exponents 4^2x-7=4^3 Even bases act as 1 2x-7=3 Ans:5
2sinx+cotx-cscx=0
Identities 2sinx+(cos/sin)-1/sinx=0 Multiply denominator to get sinx out 2sin^2x+cos-1 Identity 2(1-cos^2x)+cosx-1 2-2cos^2x+cosx-1 2cos^2-cosx-1 2u^2-u-1 (2u+1)(u-1) cosx=1/2 & cos=-1
cos2x-3sinx-2=0
Identity (1-2sin^2x)-3sinx-2=0 Simplify 2sin^2x+3sinx+1=0 Factor (2u^2+1)(u+1)
2cosθ+1=secθ
Identity 2cosθ+1=1/cosθ Put all on one side 2cosθ+1-1/cosθ=0 Multiply by denominator 2cos^2+cosθ-1=0 Factor (2u^2+1)(2u-1)
Sin2θ-Cosθ=0
Identity 2sinθcosθ-cosθ=0 Factor out cosθ cosθ(2sinθ-1)=0 Solve individually cosθ=0 sinθ=1/2
4cos^2-4sinx-5=0
Identity 4(1-sin^2x)-4sinx-5 Distribute 4-4sin^2x-4sinx-5 Combine like terms -4sin^2x-4sinx-1 4u^2+4u+1 Factor (4u+2)^2
cosx-cos2x=0
Identity cosx-1+2cos^2x=0 Factor 2Cos^2x+cosx-1=0 (2u^2+1)(u-1)=0 cosx=-1/2 & cosx=1
sin(x)+2sin(x)cos(x)=0
Identity (solve for opposite identity of independent term) sin(x)+(1+2cosx)=0 Solve for x sinx=0 Cos=-1/2
rad3secθ=2
Identity and divide 1/cos=2/rad3 Inverse/flip cosθ=rad3/2
10^x=42
Log(base10)x=42
3sinx+5=-2sinx
Set equal to 0 5sinx+5=0 solve for sin 5sinx=-5 Sinx=-5/5=-1 Sinx=-1 = 3pi/2
4sinθ-2cscθ=0
Set equal to eacother 4sinθ=2cscθ Simplify 2sinθ=cscθ Identities 2sinθ=1/sinθ Multiply both sides by sinθ 2sinθ^2=1 sinθ^2=1/2 & sinθ= (+,-)rad2/2
(sinx-1)(2sinx-1)=0
Solve for each individually Sinx-1 = sin =1 2sinx-1 = sin =1/2
sinx+cosx=rad2
Subtract one of the terms to rad2 sinx=rad2-cosx Square both sides Sin^2x=2-2rad2cosx+cos^2 Identity 1-cos^2x Set equal to 0 2cos^2-2rad2cosx+1 (rad2cos-1)^2=0 cosx=-1/rad2 = -(rad2/2)
