# §4.42 Solution of Triangles

## §4.42(i) Planar Right Triangles

 4.42.1 $\sin A=\frac{a}{c}=\frac{1}{\csc A},$ ⓘ Symbols: $\csc\NVar{z}$: cosecant function, $\sin\NVar{z}$: 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: Annotations for 4.42(i), 4.42 and 4
 4.42.2 $\cos A=\frac{b}{c}=\frac{1}{\sec A},$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\sec\NVar{z}$: 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: Annotations for 4.42(i), 4.42 and 4
 4.42.3 $\tan A=\frac{a}{b}=\frac{1}{\cot A}.$ ⓘ Symbols: $\cot\NVar{z}$: cotangent function, $\tan\NVar{z}$: 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: Annotations for 4.42(i), 4.42 and 4

## §4.42(ii) Planar Triangles

 4.42.4 $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},$ ⓘ Symbols: $\sin\NVar{z}$: sine function, $a$: length, $b$: base, $c$: length, $A$: angle, $B$: angle and $C$: angle A&S Ref: 4.3.148 Permalink: http://dlmf.nist.gov/4.42.E4 Encodings: TeX, pMML, png See also: Annotations for 4.42(ii), 4.42 and 4
 4.42.5 $c^{2}=a^{2}+b^{2}-2ab\cos C,$ ⓘ Symbols: $\cos\NVar{z}$: 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: Annotations for 4.42(ii), 4.42 and 4
 4.42.6 $a=b\cos C+c\cos B$ ⓘ Symbols: $\cos\NVar{z}$: 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: Annotations for 4.42(ii), 4.42 and 4
 4.42.7 $\hbox{area}=\tfrac{1}{2}bc\sin A=\left(s(s-a)(s-b)(s-c)\right)^{1/2},$ ⓘ Symbols: $\sin\NVar{z}$: 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: Annotations for 4.42(ii), 4.42 and 4

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

## §4.42(iii) Spherical Triangles

 4.42.8 $\cos a=\cos b\cos c+\sin b\sin c\cos A,$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $A$: angle, $a$: arc length, $b$: arc length and $c$: arc length A&S Ref: 4.3.149 Permalink: http://dlmf.nist.gov/4.42.E8 Encodings: TeX, pMML, png See also: Annotations for 4.42(iii), 4.42 and 4
 4.42.9 $\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},$ ⓘ Symbols: $\sin\NVar{z}$: sine function, $A$: angle, $B$: angle, $C$: angle, $a$: arc length, $b$: arc length and $c$: arc length A&S Ref: 4.3.149 Permalink: http://dlmf.nist.gov/4.42.E9 Encodings: TeX, pMML, png See also: Annotations for 4.42(iii), 4.42 and 4
 4.42.10 $\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$
 4.42.11 $\cos a\cos C=\sin a\cot b-\sin C\cot B,$
 4.42.12 $\cos A=-\cos B\cos C+\sin B\sin C\cos a.$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $A$: angle, $B$: angle, $C$: angle and $a$: arc length A&S Ref: 4.3.149 Permalink: http://dlmf.nist.gov/4.42.E12 Encodings: TeX, pMML, png See also: Annotations for 4.42(iii), 4.42 and 4

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