4 Elementary FunctionsApplications4.41 Sums4.43 Cubic Equations

§4.42 Solution of Triangles

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Keywords:

applications, solution of, solution of triangles and spherical triangles, triangles, trigonometric functions

Permalink:

http://dlmf.nist.gov/4.42

See also:

Annotations for Ch.4

Contents

1. §4.42(i) Planar Right Triangles
2. §4.42(ii) Planar Triangles
3. §4.42(iii) Spherical Triangles

§4.42(i) Planar Right Triangles

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Notes:

See Hobson (1928, p. 18).

Permalink:

http://dlmf.nist.gov/4.42.i

See also:

Annotations for §4.42 and Ch.4

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Defines:

$A$: angle (locally), $B$: angle (locally), $C$: right angle (locally), $a$: height (locally), $b$: base (locally) and $c$: hypotenuse (locally)

Permalink:

http://dlmf.nist.gov/4.42.F1

Encodings:

pdf, png

See also:

Annotations for §4.42(i), §4.42 and Ch.4

4.42.1		$$\sin A=\frac{a}{c}=\frac{1}{\csc A},$$
ⓘ Symbols: $\csc z$: cosecant function, $\sin z$: 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 Ch.4

4.42.2		$$\cos A=\frac{b}{c}=\frac{1}{\sec A},$$
ⓘ Symbols: $\cos z$: cosine function, $\sec z$: 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 Ch.4

4.42.3		$$\tan A=\frac{a}{b}=\frac{1}{\cot A}.$$
ⓘ Symbols: $\cot z$: cotangent function, $\tan z$: 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 Ch.4

§4.42(ii) Planar Triangles

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Notes:

See Hobson (1928, Chapter 10).

Permalink:

http://dlmf.nist.gov/4.42.ii

See also:

Annotations for §4.42 and Ch.4

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Defines:

$a$: length (locally), $b$: base (locally), $c$: length (locally), $A$: angle (locally), $B$: angle (locally) and $C$: angle (locally)

Permalink:

http://dlmf.nist.gov/4.42.F2

Encodings:

pdf, png

See also:

Annotations for §4.42(ii), §4.42 and Ch.4

4.42.4		$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},$$
ⓘ Symbols: $\sin z$: 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 Ch.4

4.42.5		$$c^2=a^2+b^2-2ab\cos C,$$
ⓘ Symbols: $\cos z$: 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 Ch.4

4.42.6		$$a=b\cos C+c\cos B$$
ⓘ Symbols: $\cos z$: 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 Ch.4

4.42.7		$$\text{area}=\frac{1}{2}bc\sin A={\left(s(s-a)(s-b)(s-c)\right)}^{1/2},$$
ⓘ Symbols: $\sin z$: 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 Ch.4

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

§4.42(iii) Spherical Triangles

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Keywords:

solution of, spherical triangles

Permalink:

http://dlmf.nist.gov/4.42.iii

See also:

Annotations for §4.42 and Ch.4

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Defines:

$A$: angle (locally), $B$: angle (locally), $C$: angle (locally), $a$: arc length (locally), $b$: arc length (locally) and $c$: arc length (locally)

Permalink:

http://dlmf.nist.gov/4.42.F3

Encodings:

pdf, png

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

4.42.8		$$\cos a=\cos b\cos c+\sin b\sin c\cos A,$$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: 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 Ch.4

4.42.9		$$\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},$$
ⓘ Symbols: $\sin z$: 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 Ch.4

4.42.10		$$\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $a$: arc length, $b$: arc length and $c$: arc length Permalink: http://dlmf.nist.gov/4.42.E10 Encodings: TeX, pMML, png See also: Annotations for §4.42(iii), §4.42 and Ch.4

4.42.11		$$\cos a\cos C=\sin a\cot b-\sin C\cot B,$$
ⓘ Symbols: $\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $B$: angle, $C$: angle, $a$: arc length and $b$: arc length Permalink: http://dlmf.nist.gov/4.42.E11 Encodings: TeX, pMML, png See also: Annotations for §4.42(iii), §4.42 and Ch.4

4.42.12		$$\cos A=-\cos B\cos C+\sin B\sin C\cos a.$$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: 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 Ch.4

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

4.41 Sums4.43 Cubic Equations

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