Showing content from https://dlmf.nist.gov/4.42 below:
§4.42 Solution of Triangles ⣠Applications ⣠Chapter 4 Elementary Functions
§4.42 Solution of Triangles â
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Keywords:
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applications, solution of, solution of triangles and spherical triangles, triangles, trigonometric functions
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http://dlmf.nist.gov/4.42
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Annotations for Ch.4
Contents
- §4.42(i) Planar Right Triangles
- §4.42(ii) Planar Triangles
- §4.42(iii) Spherical Triangles
§4.42(i) Planar Right Triangles â
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Notes:
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See Hobson (1928, p. 18).
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http://dlmf.nist.gov/4.42.i
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Annotations for §4.42 and Ch.4
Figure 4.42.1: Planar right triangle. Magnify â
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Defines:
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A : angle (locally), B : angle (locally), C : right angle (locally), a : height (locally), b : base (locally) and c : hypotenuse (locally)
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http://dlmf.nist.gov/4.42.F1
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Annotations for §4.42(i), §4.42 and Ch.4
4.42.1 sin â¡ A = a c = 1 csc â¡ A , â
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Symbols:
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csc â¡ z : cosecant function, sin â¡ z : sine function, A : angle, a : height and c : hypotenuse
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A&S Ref:
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4.3.147
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http://dlmf.nist.gov/4.42.E1
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Annotations for §4.42(i), §4.42 and Ch.4
4.42.2 cos â¡ A = b c = 1 sec â¡ A , â
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Symbols:
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cos â¡ z : cosine function, sec â¡ z : secant function, A : angle, b : base and c : hypotenuse
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A&S Ref:
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4.3.147
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http://dlmf.nist.gov/4.42.E2
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Annotations for §4.42(i), §4.42 and Ch.4
4.42.3 tan â¡ A = a b = 1 cot â¡ A . â
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Symbols:
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cot â¡ z : cotangent function, tan â¡ z : tangent function, A : angle, a : height and b : base
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A&S Ref:
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4.3.147
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http://dlmf.nist.gov/4.42.E3
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Annotations for §4.42(i), §4.42 and Ch.4
§4.42(ii) Planar Triangles â
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Notes:
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See Hobson (1928, Chapter 10).
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http://dlmf.nist.gov/4.42.ii
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Annotations for §4.42 and Ch.4
Figure 4.42.2: Planar triangle. Magnify â
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Defines:
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a : length (locally), b : base (locally), c : length (locally), A : angle (locally), B : angle (locally) and C : angle (locally)
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http://dlmf.nist.gov/4.42.F2
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Annotations for §4.42(ii), §4.42 and Ch.4
4.42.4 a sin â¡ A = b sin â¡ B = c sin â¡ C , â
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Symbols:
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sin â¡ z : sine function, a : length, b : base, c : length, A : angle, B : angle and C : angle
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A&S Ref:
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4.3.148
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http://dlmf.nist.gov/4.42.E4
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Annotations for §4.42(ii), §4.42 and Ch.4
4.42.5 c2 = a2 + b2 â 2 ⢠a ⢠b ⢠cos â¡ C , â
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Symbols:
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cos â¡ z : cosine function, a : length, b : base, c : length and C : angle
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A&S Ref:
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4.3.148
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http://dlmf.nist.gov/4.42.E5
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Annotations for §4.42(ii), §4.42 and Ch.4
4.42.6 a = b ⢠cos â¡ C + c ⢠cos â¡ B â
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Symbols:
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cos â¡ z : cosine function, a : length, b : base, c : length, B : angle and C : angle
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A&S Ref:
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4.3.148
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http://dlmf.nist.gov/4.42.E6
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Annotations for §4.42(ii), §4.42 and Ch.4
4.42.7 area = 1 2 ⢠b ⢠c ⢠sin â¡ A = ( s ⢠( s â a ) ⢠( s â b ) ⢠( s â c ) ) 1 / 2 , â
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Symbols:
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sin â¡ z : sine function, a : length, b : base, c : length, s : semi-perimeter and A : angle
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A&S Ref:
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4.3.148
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http://dlmf.nist.gov/4.42.E7
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Annotations for §4.42(ii), §4.42 and Ch.4
where s = 1 2 ⢠( a + b + c ) (the semiperimeter).
§4.42(iii) Spherical Triangles â
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Keywords:
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solution of, spherical triangles
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http://dlmf.nist.gov/4.42.iii
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See also:
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Annotations for §4.42 and Ch.4
Figure 4.42.3: Spherical triangle. Magnify â
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Defines:
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A : angle (locally), B : angle (locally), C : angle (locally), a : arc length (locally), b : arc length (locally) and c : arc length (locally)
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http://dlmf.nist.gov/4.42.F3
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pdf, png
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Annotations for §4.42(iii), §4.42 and Ch.4
4.42.8 cos â¡ a = cos â¡ b ⢠cos â¡ c + sin â¡ b ⢠sin â¡ c ⢠cos â¡ A , â
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Symbols:
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cos â¡ z : cosine function, sin â¡ z : sine function, A : angle, a : arc length, b : arc length and c : arc length
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A&S Ref:
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4.3.149
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http://dlmf.nist.gov/4.42.E8
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Annotations for §4.42(iii), §4.42 and Ch.4
4.42.9 sin â¡ A sin â¡ a = sin â¡ B sin â¡ b = sin â¡ C sin â¡ c , â
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Symbols:
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sin â¡ z : sine function, A : angle, B : angle, C : angle, a : arc length, b : arc length and c : arc length
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A&S Ref:
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4.3.149
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http://dlmf.nist.gov/4.42.E9
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Annotations for §4.42(iii), §4.42 and Ch.4
4.42.10 sin â¡ a ⢠cos â¡ B = cos â¡ b ⢠sin â¡ c â sin â¡ b ⢠cos â¡ c ⢠cos â¡ A , â
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Symbols:
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cos â¡ z : cosine function, sin â¡ z : sine function, A : angle, B : angle, a : arc length, b : arc length and c : arc length
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Permalink:
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http://dlmf.nist.gov/4.42.E10
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Annotations for §4.42(iii), §4.42 and Ch.4
4.42.11 cos â¡ a ⢠cos â¡ C = sin â¡ a ⢠cot â¡ b â sin â¡ C ⢠cot â¡ B , â
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Symbols:
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cos â¡ z : cosine function, cot â¡ z : cotangent function, sin â¡ z : sine function, B : angle, C : angle, a : arc length and b : arc length
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Permalink:
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http://dlmf.nist.gov/4.42.E11
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Annotations for §4.42(iii), §4.42 and Ch.4
4.42.12 cos â¡ A = â cos â¡ B ⢠cos â¡ C + sin â¡ B ⢠sin â¡ C ⢠cos â¡ a . â
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Symbols:
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cos â¡ z : cosine function, sin â¡ z : sine function, A : angle, B : angle, C : angle and a : arc length
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A&S Ref:
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4.3.149
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Permalink:
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http://dlmf.nist.gov/4.42.E12
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Encodings:
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TeX, pMML, png
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Annotations for §4.42(iii), §4.42 and Ch.4
For these and other formulas see Smart (1962, Chapter 1).
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