Comprehensive Study Guide on Hyperbolic Functions, Series Summation, and Spherical Trigonometry

Definitions and Basic Properties of Hyperbolic Functions

• Foundational Definitions: Hyperbolic functions describe relationships based on the exponential constant $e$. The Swiss mathematician Johann Heinrich Lambert (1728-1777) conducted the first systematic consideration of these functions.

• Primary Hyperbolic Functions:

  • Hyperbolic Cosine: Defined for all real values of $x$ as $\cosh(x) = \frac{1}{2}(e^x + e^{-x})$.
  • Hyperbolic Sine: Defined for all real values of $x$ as $\sinh(x) = \frac{1}{2}(e^x - e^{-x})$.

• Secondary Hyperbolic Functions:

  • Hyperbolic Tangent: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$.
  • Hyperbolic Cotangent: $\coth(x) = \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$.
  • Hyperbolic Secant: $\text{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}$.
  • Hyperbolic Cosecant: $\text{cosech}(x) = \frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}}$.

• Exponential Series Expansions:

  • $e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
  • $e^{-x} = 1 - \frac{x}{1!} + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots$
  • $\cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots$
  • $\sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \dots$

• Periodicity: Hyperbolic sine and cosine are periodic functions with an imaginary period of $2\pi i$.

  • $\sinh(x + 2n\pi i) = \sinh(x)$
  • $\cosh(x + 2n\pi i) = \cosh(x)$

Relations Between Hyperbolic and Circular Functions

• Imaginary Argument Relations:

  • $\sin(ix) = i\sinh(x)$
  • $\cos(ix) = \cosh(x)$
  • $\tan(ix) = i\tanh(x)$
  • $\sinh(ix) = i\sin(x)$
  • $\cosh(ix) = \cos(x)$
  • $\tanh(ix) = i\tan(x)$

• Inverse Relations:

  • $\sinh^{-1}(z) = -i\sin^{-1}(iz)$
  • $\cosh^{-1}(z) = -i\cos^{-1}(z)$, though sometimes expressed as $\cosh^{-1}(x) = \ln(x \pm \sqrt{x^2-1})$.
  • $\tanh^{-1}(z) = -i\tan^{-1}(iz)$.

Hyperbolic Identities and Osborn's Rule

• Osborn's Rule: This rule transforms trigonometric identities into hyperbolic identities. Replace $\cos$ with $\cosh$ and $\sin$ with $\sinh$. If there is a product of two expressions involving sine (e.g., $\sin^2(x)$ or $\sin(A)\sin(B)$), a sign change must be applied.

  • Example: $\cos(2x) = 1 - 2\sin^2(x)$ becomes $\cosh(2x) = 1 + 2\sinh^2(x)$.
  • Example: $\sec^2(x) = 1 + \tan^2(x)$ involves a ratio involving $\sinh(x)$, thus it becomes $\text{sech}^2(x) = 1 - \tanh^2(x)$.

• Fundamental Identities:

  • $\cosh^2(x) - \sinh^2(x) = 1$
  • $\text{sech}^2(x) + \tanh^2(x) = 1$
  • $\coth^2(x) - \text{cosech}^2(x) = 1$
  • $\cosh(x) + \sinh(x) = e^x$
  • $\cosh(x) - \sinh(x) = e^{-x}$

• Addition and Double Angle Formulas:

  • $\sinh(x \pm y) = \sinh(x)\cosh(y) \pm \cosh(x)\sinh(y)$
  • $\cosh(x \pm y) = \cosh(x)\cosh(y) \pm \sinh(x)\sinh(y)$
  • $\tanh(x \pm y) = \frac{\tanh(x) \pm \tanh(y)}{1 \pm \tanh(x)\tanh(y)}$
  • $\sinh(2x) = 2\sinh(x)\cosh(x)$
  • $\cosh(2x) = \cosh^2(x) + \sinh^2(x) = 2\cosh^2(x) - 1 = 1 + 2\sinh^2(x)$
  • $\tanh(2x) = \frac{2\tanh(x)}{1 + \tanh^2(x)}$

Graphs and Solving Equations

• Graph of $\cosh(x)$: Found in nature as the "catenary," the shape assumed by a chain hanging freely between two supports. The function has a domain of $x \in \mathbb{R}$ and a range of $y \ge 1$.
• Graph of $\sinh(x)$: Defined for all real $x$, passing through the origin with a gradient of $1$. The range is all real numbers.
• Graph of $\tanh(x)$: Domain of all real numbers; range is the open interval $(-1, 1)$.
• Solving Equations:
  • Equations such as $2\cosh(2x) + 10\sinh(2x) = 5$ can be solved by substituting exponential definitions: $e^{2x} + e^{-2x} + 5(e^{2x} - e^{-2x}) = 5$.
  • Simplifying leads to a quadratic in $e^{2x}$: $6e^{4x} - 5e^{2x} - 4 = 0$.
  • Factoring $(3e^{2x} - 4)(2e^{2x} + 1) = 0$ gives one real solution: $e^{2x} = \frac{4}{3}$, hence $x = \frac{1}{2}\ln(\frac{4}{3})$.

Calculus of Hyperbolic Functions

• Derivatives:

  • $\frac{d}{dx}(\cosh(x)) = \sinh(x)$
  • $\frac{d}{dx}(\sinh(x)) = \cosh(x)$
  • $\frac{d}{dx}(\tanh(x)) = \text{sech}^2(x)$
  • $\frac{d}{dx}(\text{sech}(x)) = -\text{sech}(x)\tanh(x)$
  • $\frac{d}{dx}(\text{cosech}(x)) = -\text{cosech}(x)\coth(x)$
  • $\frac{d}{dx}(\coth(x)) = -\text{cosech}^2(x)$

• Standard Integrals:

  • $\int \cosh(ax)\,dx = \frac{1}{a}\sinh(ax) + C$
  • $\int \sinh(ax)\,dx = \frac{1}{a}\cosh(ax) + C$
  • $\int \sinh^2(x)\,dx = \int \frac{1}{2}(\cosh(2x) - 1)\,dx = \frac{1}{4}\sinh(2x) - \frac{1}{2}x + C$
  • $\int x\sinh(x)\,dx = x\cosh(x) - \sinh(x) + C$

Inverse Hyperbolic Functions

• Definitions and Domains:

  • $\sinh^{-1}(x)$: Domain $x \in \mathbb{R}$, range $y \in \mathbb{R}$.
  • $\cosh^{-1}(x)$: Principal value defined for domain $x \ge 1$, range $y \ge 0$.
  • $\tanh^{-1}(x)$: Domain $|x| < 1$, range $y \in \mathbb{R}$.

• Logarithmic Equivalents:

  • $\sinh^{-1}(x) = \ln(x + \sqrt{x^2+1})$ for all $x$.
  • $\cosh^{-1}(x) = \ln(x + \sqrt{x^2-1})$ for $x \ge 1$.
  • $\tanh^{-1}(x) = \frac{1}{2}\ln(\frac{1+x}{1-x})$ for $|x| < 1$.

• Derivatives of Inverse Functions:

  • $\frac{d}{dx}(\sinh^{-1}(x)) = \frac{1}{\sqrt{x^2+1}}$
  • $\frac{d}{dx}(\cosh^{-1}(x)) = \frac{1}{\sqrt{x^2-1}}$
  • $\frac{d}{dx}(\tanh^{-1}(x)) = \frac{1}{1-x^2}$

Hyperbolic Substitutions in Integration

• Forms for $\sqrt{x^2+a^2}$: Use $x = a\sinh(u)$.
  • Example: $\int \frac{1}{\sqrt{x^2+a^2}}\,dx = \sinh^{-1}(\frac{x}{a}) + C$.
• Forms for $\sqrt{x^2-a^2}$: Use $x = a\cosh(u)$.
  • Example: $\int \frac{1}{\sqrt{x^2-a^2}}\,dx = \cosh^{-1}(\frac{x}{a}) + C$.
• Completing the Square: For integrals involving quadratic denominators like $\int \frac{1}{\sqrt{4x^2 - 8x - 5}}\,dx$, rewriting as $\int \frac{1}{\sqrt{4(x-1)^2 - 9}}\,dx$ allows substitution $2(x-1) = 3\cosh(u)$.

Summation of Trigonometric Series

• Method of Difference: Used for series where the $r$-th term $t_r$ can be expressed as $f(r+1) - f(r)$.
  • Example: $\sum_{R=1}^{n} \tan^{-1}(\frac{2}{2R^2})$. Using logic $t_R = \tan^{-1}(2R+1) - \tan^{-1}(2R-1)$, the sum telescopically collapses to $S_n = \tan^{-1}(2n+1) - \tan^{-1}(1)$.
• Sum of Sines in Arithmetic Progression (A.P.):
  • For angles $\alpha, \alpha+\beta, \alpha+2\beta, \dots, \alpha+(n-1)\beta$, the sum is $\frac{\sin(\frac{n\beta}{2})}{\sin(\frac{\beta}{2})} \times \sin(\alpha + \frac{n-1}{2}\beta)$.
• C+iS Method: Represents a series as the real part ($C$) and imaginary part ($S$) of a complex power series.
  • Example: For $C = \sum_{n=1}^{\infty} \cos\theta\cos\theta + \cos^2\theta\cos(2\theta) + \dots$, we use $C+iS = \cos\theta e^{i\theta}(1 - \cos\theta e^{i\theta})^{-1}$. Solving leads to $C=0$.

Introduction to Spherical Geometry

• The Sphere: A curved surface where every point is equidistant from a fixed interior point (the centre). A plane section of a sphere is always a circle.
• Great Circles: Circles formed by planes passing through the centre of the sphere. Their radius equals the radius of the sphere. They form the shortest path between two points on the surface.
• Small Circles: Sections formed by planes that do not pass through the centre.
• Axis and Poles: The axis of a circle is the sphere's diameter perpendicular to the circle's plane. The extremities are the poles. Poles of great circles are always $90^\circ$ (quadrants) from the circumference.
• Secondaries: Great circles passing through the poles of another great circle are called secondaries to that circle.

Spherical Trigonometry and Triangles

• Spherical Triangle: A figure on a sphere bounded by three arcs of great circles.
• Properties of Angles:
  • Each angle is less than $\pi$.
  • The sum of the angles $\alpha, \beta, \gamma$ satisfies $\pi < \alpha + \beta + \gamma < 3\pi$.
• Properties of Sides:
  • The sum of sides $a+b+c < 360^\circ$ ($2\pi$).
  • Any single side is less than the sum of the other two ($a < b+c$).
• Polar Triangle: Defined by vertices $A', B', C'$ which are the poles of the sides $BC, CA, AB$ of the triangle $ABC$.

Formulae for Spherical Triangles

• Cosine Formula (Three sides and one angle):
  • $\cos(A) = \frac{\cos(a) - \cos(b)\cos(c)}{\sin(b)\sin(c)}$
  • Alternately written: $\cos(a) = \cos(b)\cos(c) + \sin(b)\sin(c)\cos(A)$.
• Sine Formula (Two sides and opposite angles):
  • $\frac{\sin(A)}{\sin(a)} = \frac{\sin(B)}{\sin(b)} = \frac{\sin(C)}{\sin(c)}$
• Supplemental Cosine Formula:
  • $\cos(a) = \frac{\cos(A) + \cos(B)\cos(C)}{\sin(B)\sin(C)}$
• Numerical Examples:
  • To find side $b$ in a triangle where $a=76^\circ, c=58^\circ, B=117^\circ$:
    • $\cos(b) = \cos(76^\circ)\cos(58^\circ) + \sin(76^\circ)\sin(58^\circ)\cos(117^\circ) \approx -0.24537$.
    • $b = \cos^{-1}(-0.24537) \approx 104.2037^\circ$.

Practice Problems and Specific Applications

• Complex Forms: To express $\cosh(x+iy)$ in $a+ib$ form: $\cosh(x+iy) = \cosh(x)\cos(y) + i\sinh(x)\sin(y)$.
• Tri-rectangular Triangles: A spherical triangle with three right angles ($90^\circ$) and three right sides is called an octant or an octant triangle.
• Stationary Values: Finding the stationary point of $f(x) = (1+x)\sinh(3(x-2))$ involves identifying the intersection of $y = \tanh(3(x-2))$ and the line $y+3x+3=0$, often using iterative techniques like Newton-Raphson starting at an estimate like $-1$.