# §4.42 Solution of Triangles

## §4.42(i) Planar Right Triangles

 4.42.1 $\mathop{\sin\/}\nolimits A=\frac{a}{c}=\frac{1}{\mathop{\csc\/}\nolimits A},$ Symbols: $\mathop{\csc\/}\nolimits\NVar{z}$: cosecant function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $A$: angle, $a$: height and $c$: hypotenuse A&S Ref: 4.3.147 Permalink: http://dlmf.nist.gov/4.42.E1 Encodings: TeX, pMML, png See also: info for 4.42(i)
 4.42.2 $\mathop{\cos\/}\nolimits A=\frac{b}{c}=\frac{1}{\mathop{\sec\/}\nolimits A},$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sec\/}\nolimits\NVar{z}$: secant function, $A$: angle, $b$: base and $c$: hypotenuse A&S Ref: 4.3.147 Permalink: http://dlmf.nist.gov/4.42.E2 Encodings: TeX, pMML, png See also: info for 4.42(i)
 4.42.3 $\mathop{\tan\/}\nolimits A=\frac{a}{b}=\frac{1}{\mathop{\cot\/}\nolimits A}.$ Symbols: $\mathop{\cot\/}\nolimits\NVar{z}$: cotangent function, $\mathop{\tan\/}\nolimits\NVar{z}$: tangent function, $A$: angle, $a$: height and $b$: base A&S Ref: 4.3.147 Permalink: http://dlmf.nist.gov/4.42.E3 Encodings: TeX, pMML, png See also: info for 4.42(i)

## §4.42(ii) Planar Triangles

 4.42.4 $\frac{a}{\mathop{\sin\/}\nolimits A}=\frac{b}{\mathop{\sin\/}\nolimits B}=% \frac{c}{\mathop{\sin\/}\nolimits C},$
 4.42.5 $c^{2}=a^{2}+b^{2}-2ab\mathop{\cos\/}\nolimits C,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $a$: length, $b$: base, $c$: length and $C$: angle A&S Ref: 4.3.148 Permalink: http://dlmf.nist.gov/4.42.E5 Encodings: TeX, pMML, png See also: info for 4.42(ii)
 4.42.6 $a=b\mathop{\cos\/}\nolimits C+c\mathop{\cos\/}\nolimits B$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $a$: length, $b$: base, $c$: length, $B$: angle and $C$: angle A&S Ref: 4.3.148 Permalink: http://dlmf.nist.gov/4.42.E6 Encodings: TeX, pMML, png See also: info for 4.42(ii)
 4.42.7 $\hbox{area}=\tfrac{1}{2}bc\mathop{\sin\/}\nolimits A=\left(s(s-a)(s-b)(s-c)% \right)^{1/2},$ Symbols: $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $a$: length, $b$: base, $c$: length, $s$: semi-perimeter and $A$: angle A&S Ref: 4.3.148 Permalink: http://dlmf.nist.gov/4.42.E7 Encodings: TeX, pMML, png See also: info for 4.42(ii)

where $s=\tfrac{1}{2}(a+b+c)$ (the semiperimeter).

## §4.42(iii) Spherical Triangles

 4.42.8 $\mathop{\cos\/}\nolimits a=\mathop{\cos\/}\nolimits b\mathop{\cos\/}\nolimits c% +\mathop{\sin\/}\nolimits b\mathop{\sin\/}\nolimits c\mathop{\cos\/}\nolimits A,$
 4.42.9 $\frac{\mathop{\sin\/}\nolimits A}{\mathop{\sin\/}\nolimits a}=\frac{\mathop{% \sin\/}\nolimits B}{\mathop{\sin\/}\nolimits b}=\frac{\mathop{\sin\/}\nolimits C% }{\mathop{\sin\/}\nolimits c},$
 4.42.10 $\mathop{\sin\/}\nolimits a\mathop{\cos\/}\nolimits B=\mathop{\cos\/}\nolimits b% \mathop{\sin\/}\nolimits c-\mathop{\sin\/}\nolimits b\mathop{\cos\/}\nolimits c% \mathop{\cos\/}\nolimits A,$
 4.42.11 $\mathop{\cos\/}\nolimits a\mathop{\cos\/}\nolimits C=\mathop{\sin\/}\nolimits a% \mathop{\cot\/}\nolimits b-\mathop{\sin\/}\nolimits C\mathop{\cot\/}\nolimits B,$
 4.42.12 $\mathop{\cos\/}\nolimits A=-\mathop{\cos\/}\nolimits B\mathop{\cos\/}\nolimits C% +\mathop{\sin\/}\nolimits B\mathop{\sin\/}\nolimits C\mathop{\cos\/}\nolimits a.$

For these and other formulas see Smart (1962, Chapter 1).