Spherical Trigonometry

Excerpts

• Spherical Trigonometry

  By $i^\frac{\pi}{2}j^\frac{\pi}{2}$ is meant a quadrant round $i$ followed by a quadrant round $j$; it is equivalent to the quadrant from $j$ to $i$, that is, to $-k^\frac{\pi}{2}$.

• Spherical Trigonometry

  This quadrantal version can be decomposed into the three rectangular components $li^\frac{\pi}{2}$, $mj^\frac{\pi}{2}$, $nk^\frac{\pi}{2}$; and these components are not successive versions, but the parts of one version.

• Spherical Trigonometry

  where $A$ denotes the external angle instead of the angle included by the sides.

• Spherical Trigonometry

  The second term is the directed sine of the angle; for the square of (2) is equal to 1 minus the square of (1), and its direction is normal to the plane of the product angle.

• Spherical Trigonometry

  In the present theory of diplanar quaternions we cannot expect to find that the sum of the logarithms of any two proposed factors shall be generally equal to the logarithm of the product;

• Spherical Trigonometry

  where the coefficients are those of the binomial theorem, the only difference being that $\cos \beta\gamma$ occurs in all the odd terms as a factor.

Equations

• Spherical Trigonometry

  \cos a = \cos b \cos c + \sin b \sin c \cos A

  The cosine of a side of a spherical triangle equals the product of the cosines of the other two sides plus the product of their sines times the cosine of the external angle opposite.

• Spherical Trigonometry

  \beta^\frac{\pi}{2}\gamma^\frac{\pi}{2} = -\cos \beta\gamma -\sin \beta\gamma \cdot \overline{\beta\gamma}^\frac{\pi}{2}

  The product of two quadrantal versors about the axes beta and gamma is a minus cosine of the product angle plus a directed sine term along the axis of the product.

• Spherical Trigonometry

  \beta^b = \cos b + \sin b \cdot \beta^\frac{\pi}{2}

  A spherical versor of ratio b about axis beta is cosine b plus sine b times the quadrantal versor about beta.

• Spherical Trigonometry

  \gamma^c = \cos c + \sin c \cdot \gamma^\frac{\pi}{2}

  A spherical versor of ratio c about axis gamma is cosine c plus sine c times the quadrantal versor about gamma.

• Spherical Trigonometry

  \cos\beta^b\gamma^c = \cos b\cos c - \sin b\sin c\cos \beta\gamma

  The cosine of the product versor beta^b gamma^c equals cos b cos c minus sin b sin c times the cosine of the product angle.

• Spherical Trigonometry

  \cos b = 1 - \frac{b^2}{2!} + \frac{b^4}{4!} - \frac{b^6}{6!} + \text{ etc.}

  The cosine of b is expanded as an infinite power series in b.

• Spherical Trigonometry

  \sin b = b - \frac{b^3}{3!} + \frac{b^5}{5!} - \text{ etc.}

  The sine of b is expanded as an infinite power series in b.

• Spherical Trigonometry

  e^{b\beta^\frac{\pi}{2}} e^{c\gamma^\frac{\pi}{2}} = e^{b\beta^\frac{\pi}{2} + c\gamma^\frac{\pi}{2}}

  The exponential of a successive sum of two quadrantal versor terms is the product of their exponentials, provided the order of the terms is preserved.

• Spherical Trigonometry

  \left\{ b \cdot \beta^\frac{\pi}{2} + c \cdot \gamma^\frac{\pi}{2} \right\}^n = b^n \cdot \beta^{n^\frac{\pi}{2}} + nb^{n-1}c \cdot \beta^{(n-1)(\frac{\pi}{2})} \gamma^\frac{\pi}{2} + \frac{n(n-1)}{1 \cdot 2} b^{n-2} c^2 \cdot \beta^{(n-2)(\frac{\pi}{2})} \gamma^\pi + \text{etc.}

  The n-th power of a successive binomial in quadrantal versors expands with binomial coefficients, keeping the order of the terms.

• Spherical Trigonometry

  \sin\alpha\beta \sin\overline{\alpha\beta}\gamma \cdot \overline{\overline{\alpha\beta}\gamma} = \cos\alpha\gamma \cdot \beta - \cos\beta\gamma \cdot \alpha

  A directed-sine product of successive axes equals the difference of two cosine-weighted axes.

• Spherical Trigonometry

  -(ll' + mm' + nn') -(mn' - m'n)i^\frac{\pi}{2} - (nl' - n'l)j^\frac{\pi}{2} -(lm' - l'm)k^\frac{\pi}{2}

  The product of two quadrantal versors about general axes resolves into a scalar part and a quadrantal part along the three coordinate axes.

• Spherical Trigonometry

  j^\frac{\pi}{2}i^\frac{\pi}{2} = k^\frac{\pi}{2}

  The product of the quadrantal versor about j followed by that about i equals the quadrantal versor about k.

• Spherical Trigonometry

  i^\frac{\pi}{2}i^\frac{\pi}{2} = -

  The square of a quadrantal versor about one of the perpendicular axes is a half-turn, written as minus one.

• Spherical Trigonometry

  y^{-c} = \cos c - \sin c \cdot \gamma^\frac{\pi}{2}

  The reciprocal of the spherical versor gamma^c is cosine c minus sine c times the quadrantal versor about gamma.

Problems