Mathematical Formulas for Industrial and Mechanical Engineering

sin α 5

a

c

5 cos β

cos α 5

b

c

5 sin β

tan α 5

a

b

5 cot β

cot α 5

b

a

5 tan β

sec α 5

c

b

5 cosec β

cosec α 5

c

a

5 sec β

4.4 Any Triangle

In any triangle with sides a, b,andc and corresponding opposite angles α, β,andγ:

b

c

a

α

β

γ

α 1 β 1 γ 5 180



a 1 b . c

b 1 c . a

a 1 c . a

The law of sines is

a

sin α

5

b

sin β

5

c

sin γ

or

sin α

a

5

sin β

b

5

sin γ

c

The law of cosines is

a

2

5 b

2

1 c

2

2 2cb cos α

b

2

5 a

2

1 c

2

2 2ca cos β

c

2

5 b

2

1 a

2

2 2ab cos γ

The law of tangents is

a 1 b

a 2 b

5

tanððα 1 βÞ=2Þ

tanððα 2 βÞ=2Þ

55Analytic Geometry and Trigonometry