反双曲函数基本公式 \sinh^ {-1} x \pm \sinh^ {-x} y = \sinh^ {-1} \left ( \sinh^ {-1}x \sqrt {1+y^2} \pm y \sqrt {1 +x^2} \right) \\\cosh ^ {-1} x \pm \cosh ^ {-1} y = \cosh^ {-1} \left [ xy \pm \sqrt { (x^2 -1) (y^2 -1) } \right] \\\tanh ^ {-1}x \pm \tanh^ {-1}y = \tanh^ {-1} \frac {x \pm y} {1 \pm xy} \\

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.

2016年5月4日 · How do you show (cosh x + sinh x)n = cosh(nx) + sinh(nx) for any real number n? Use the definition coshx = ex + e−x 2 and sinhx = ex −e−x 2. Left Side: [ex +e−x 2 + ex − e−x …

Hyperbolic Functions - sinh, cosh, tanh, coth, sech, csch Definition of hyperbolic functions Hyperbolic sine of x \displaystyle \text {sinh}\ x = \frac {e^ {x} - e^ {-x}} {2} sinh x = 2ex −e−x …

2016年4月28日 · You can easily generalize this to show that if n, m n, m are positive integers and cosh(nx) cosh (n x) and cosh((n + m)x) cosh ((n + m) x) are rational, then cosh(mx) cosh (m x) is rational

Hyperbolic Functions The two basic hyperbolic functions are "sinh" and "cosh": Hyperbolic Sine: sinh (x) = ex − e-x 2 (pronounced "shine") Hyperbolic Cosine: cosh (x) = ex + e-x 2 (pronounced "cosh") They use the natural exponential function ex And are not the same as sin (x) and cos (x), but a little bit similar: sinh vs sin cosh vs cos Catenary

这里介绍一些有关 双曲三角函数 的恒等式，它们是 三角函数 的 三角恒等式 推广。 归一恒等式. {\displaystyle \cosh^2 x - \sinh^2 x = 1, \quad \tanh^2 x + \operatorname {sech}^2 x = 1, \quad \coth^2 x - 1 = \operatorname {csch}^2 x.} 欧拉公式. e {\displaystyle \text {e}^ {x} = \cosh x + \sinh x, \quad \text {e}^ {-x} = \cosh x - \sinh x.} 双曲三角函数的相互表示可以根据基本恒等式和它们之间的相互定义来推出。

2024年7月4日 · 该代码使用Sympy库计算了曲线y = cosh (x)在x轴和两条竖直线x = a和x = b之间的面积。 Sympy的integrate ()函数用于计算积分。 cosh函数在求解一阶微分方程中具有重要作用。 考虑以下一阶微分方程： 其中 a 和 b 为常数。 然后，使用积分因子法求解该方程。 积分因子为： 其中 C 为积分常数。 cosh函数在求解二阶微分方程中也有应用。 考虑以下二阶微分方程： 其中 a 和 b 为常数。 我们可以使用特征方程法求解该方程。 特征方程为： 其中 c1 和 c2 为积分常数 …

2020年12月7日 · Prove that (cosh (x) + sinh (x))^n = cosh (nx) + sinh (nx)If you enjoyed this video please consider liking, sharing, and subscribing.Udemy Courses Via My Website...

2017年10月16日 · Prove that (cosh x + sinh x)n = cosh nx + sinh nx? Please see below. As coshx = ex + e−x 2 and sinhx = ex −e−x 2. and adding two coshx +sinhx = ex. and (coshx + sinhx)n = (ex)n = enx. also coshnx = enx + e−nx 2 and sinhnx = enx −e−nx 2. and coshnx + sinhnx = enx. Hence, (coshx + sinhx)n = coshnx + sinhnx. Proof by mathematical induction.