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# Find value of R and a such that: 5coshx-4sinhx=Rcosh(x-a)?

### 1 Answer

- 2 months agoFavorite Answer
cosh(t) = (1/2) * (e^(t) + e^(-t))

sinh(t) = (1/2) * (e^(t) - e^(-t))

5 * cosh(x) - 4 * sinh(x) =>

5 * (1/2) * (e^(x) + e^(-x)) - 4 * (1/2) * (e^(x) - e^(-x)) =>

(1/2) * (5 * e^(x) - 4 * e^(x) + 5 * e^(-x) + 4 * e^(-x)) =>

(1/2) * (e^(x) + 9 * e^(-x))

r * cosh(x - a) = r * (1/2) * (e^(x - a) + e^(a - x))

r * (1/2) * (e^(x - a) + e^(a - x)) = (1/2) * (e^(x) + 9 * e^(-x))

r * e^(x - a) + r * e^(a - x) = e^(x) + 9 * e^(-x)

r * e^(x - a) = e^(x)

r * e^(x) / e^(a) = e^(x)

r / e^(a) = 1

r * e^(a - x) = 9 * e^(-x)

r * e^(a) / e^(x) = 9 / e^(x)

r * e^(a) = 9

r = e^(a)

r * e^(a) = 9

r * r = 9

r^2 = 9

r = +/- 3

Since e^(a) > 0 for all real values of a, r > 0 as well (in order for us to have coefficients like 1 and 9, it has to be positive as well)

r = 3

3 / e^(a) = 1

3 = e^(a)

ln(3) = a