University:
Kentucky State University
Course:
MAT 120 | Precalculus
Academic year:

2019
Views:

383

Pages:

5
Author:

Athena Harrison

**Hyperbolic Function** * **Hyperbolic Sine:** sinh x = (e^x - e^-x)/2 * **Hyperbolic Cosine:** cosh x = (e^x + e^-x)/2 * **Hyperbolic Tangent:** tanh x = sinh x / cosh x = (e^x - e^-x) / (e^x + e^-x) * **Hyperbolic Cotangent:** coth x = cosh x / sinh x = (e^x + e^-x) / (e^x - e^-x) * **Hyperbolic Secant:** sech x = 1 / cosh x = 2 / (e^x + e^-x) * **Hyperbolic Cosecant:** csch x = 1 / sinh x = 2 / (e^x - e^-x) **Evaluate:** 1. cosh(0) = (e^0 + e^0)/2 = 1 2. sinh(0) = (e^0 - e^0)/2 = 0 3. cosh(ln2) = (e^(ln2) + e^(-ln2))/2 = (2 + 1/2)/2 = 5/4 4. cosh(2lnx) = cosh(lnx²) = (e^(lnx²) + e^(-lnx²))/2 = (x² + 1/x²)/2 = (x^4 + 1)/(2x²) 5. sinh(lnx - 3lnx²) = sinh(ln(x/x³)) = sinh(ln(1/x²)) = (e^(ln(1/x²)) - e^(-ln(1/x²)))/2 = (1/x² - x²)/2 **Hyperbolic Functions** 1. cosh(x) + sinh(x) = e^x 2. cosh(x) - sinh(x) = e^-x 3. cosh²(x) - sinh²(x) = 1 4. 1 - tanh²(x) = sech²(x) 5. coth²(x) - 1 = csch²(x) 6. cosh(2x) = cosh²(x) + sinh²(x) 7. sinh(2x) = 2sinh(x)cosh(x) 8. cosh(-x) = cosh(x) 9. sinh(-x) = -sinh(x) **Solve for x:** 1. cosh(x) - sinh(x) = 6 e^-x = 6 => -x = ln(6) => x = -ln(6) 2. sinh(x) = 2 (e^x - e^-x)/2 = 2 => e^x - e^-x = 4 e^x - 4 - 1/e^x = 0 e^(2x) - 4e^x - 1 = 0 (e^x - 2)^2 - 5 = 0 e^x = 2 ± √5 ln(e^x) = ln(2 ± √5) => x = ln(2 ± √5) **Graphs:** * **cosh(x):** * Domain: R * Range: [1, ∞) * Even function (symmetric about y-axis) * cosh(0) = 1 * lim(x→∞) cosh(x) = ∞ * **sinh(x):** * Domain: R * Range: R * Odd function (symmetric about origin) * sinh(0) = 0 * lim(x→∞) sinh(x) = ∞ * lim(x→-∞) sinh(x) = -∞ **Hyperbolic Functions** **y = tanh x** * tanh x = (e^x - e^-x) / (e^x + e^-x) * lim(x→∞) tanh x = 1 * lim(x→-∞) tanh x = -1 * HA: y = ±1 * Odd **y = coth x** * coth x = (e^x + e^-x) / (e^x - e^-x) * lim(x→∞) coth x = 1 * lim(x→-∞) coth x = -1 * HA: y = ±1 * Odd **y = sech x** * sech x = 2 / (e^x + e^-x) * lim(x→±∞) sech x = 0 * HA: y = 0 * Even **y = csch x** * csch x = 2 / (e^x - e^-x) * lim(x→∞) csch x = 0 * lim(x→-∞) csch x = 0 * Have vertical asymptote