Karteikarten
Hyperbolic Identities
Karten in diesem Stapel (7)
- 1tanh x ≡ .......sinhx / coshx
- 2cosh^2x - sinh^2x ≡ ....1
(Extra note on the - symbol!!) - 3In log form, solve: cosh^2x = 7 - sinhxcos^2x = 1+sinh^2x , so we can say:
sinh^2x + sinhx - 6 = 0
Solve this quadratic for sinhx = -3 or 2
Using 3: sinhx-1(3) can be written exponentially as ln(-3+root(9+1))
(Same for 2) - 4In log form, solve: coshx + 4sinhx = 0As = 0 , divide through by coshx:
coshx/coshx + 4sinhx/coshx = 0
1+4tanhx=6 , same as tanhx=-1/4
tanhx-1(-1/4) = x , x=0.5 x ln(3/5) - 5In log form, solve: coshx + 2sinhx = 3Has we have a number, we need to solve this using exponentials:
(e^x + e^-x)/2 + 2(e^x - e^-x)/2 = 3
Add up for quadratic: 3e^x - e^-x -6=0
Multiply through by e^x to get rid of e^-x
3(e^x)^2 - 6e^x -1 = 0 , solve for:
e^x = (3+/-2root(3)/3). We know (3-2root(3)/3) has no solutions, so: ln(3+2root(3)/3) - 6When solving coshx = a , we must remember.....There is a negative solution as well
- 7When can work out the other value of coshx by.....x = +/-ln(a+root(a^2-1))
OR
x = ln(a+/-root(a^2-1))