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DLMF: §4.42 Solution of Triangles ‣ Applications ‣ Chapter 4 Elementary Functions

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4 Elementary Functions Applications 4.41 Sums 4.43 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

§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

See accompanying text — Figure 4.42.1: Planar right triangle. Magnify

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 Ch.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 Ch.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 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

See accompanying text — Figure 4.42.2: Planar triangle. Magnify

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 Ch.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 Ch.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 Ch.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 Ch.4

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

See accompanying text — Figure 4.42.3: Spherical triangle. Magnify

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 Ch.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 Ch.4

4.42.10		$\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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: $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\sin\NVar{z}$ : 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: $\cos\NVar{z}$ : 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 Ch.4

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

4.41 Sums 4.43 Cubic Equations

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