Plane Geometry, Plane Figures (Geometric Figures) - Triangles Congruence The sine law (or the sine rule) and the cosine law
The sine law
The sine law examples
Solving the oblique triangle - use of the sine law and the cosine law
Congruence
Two triangles are congruent if they have identical size and shape so that they can be exactly superimposed.
Thus, two triangles are congruent:
a) if a pair of corresponding sides and the included angle are equal,
b) if their corresponding sides are equal,
c) if a pair of corresponding angles and the included side are equal.
The sine law (or the sine rule) and the cosine law
From the congruence of triangles follows that an oblique triangle is determined by three of its parts, as are
- two sides and the included angle (SAS),
- two angles and the included side (ASA),
- three sides (SSS) and
- two sides and the angle opposite one of them (SSA), which does not always determine a unique triangle.
By using definitions of  trigonometric functions of an acute angle and Pythagoras’ theorem, we can examine mutual relationships of sides and corresponding angles of an oblique triangle.
The sine law
From the right triangles, ACD and BCD in the figure,
hc = a sinb   and   hc = b sina,
so that,               a sinb = b sina,
 therefore,  Expressing the same way the altitudes, hb and ha as common legs of another pairs of two right triangles in the given triangle, we get
hb = a sing   and   hb = c sina,
so that,            a sing = c sina,
 or and,     ha = b sing   and   ha = c sinb,
so that,            b sing = c sinb,
 or These relations are called the sine law and in words:
Sides of a triangle are to one another in the same ratio as sine of the corresponding (opposite) angles.
As these ratios express relations of any of two sides and their opposite angles, it follows that the sine law can be applied to solve an oblique triangle in the cases when given are,
- two angles and the included side (ASA) and
- two sides and the angle opposite one of them (SSA), that does not specify a triangle uniquely.
Example:   In the oblique triangle ABC side a = 6 cm, angles, a = 38° and g = 120°, find the remaining sides b and c and angle b.
Solution:  Given a = 6 cm, a = 38° and g = 120°.    b, c and b = ?
As  a + b + g = 180°    then     b = 180° - (a + g) = 180° - 158° = 22°,  and since  Example:   In the oblique triangle with sides, b = 7 cm and c = 4 cm and angle b = 115°, find the side a and angles, a and g.
Solution:  Given b = 7 cm, c = 4 cm and b = 115°.    a, a and g = ?
Sequence of calculating the particular part of the triangle depends of the given parts, that is, use the ratio that includes all three known parts and the one needed to be calculated, so   Example:   In the triangle with sides, b = 7 cm and c = 4 cm and angle g = 31°11′27″, find the side a and angles, a and b.
Solution:  Given b = 7 cm, c = 4 cm and g = 31°11′27″.    a, a and b = ?
As shows the below figure it is possible construct two triangles with given parts, one acute, the other obtuse.
That is, the problem has two solutions, one triangle with angle b and the other with angle b = 180° - b.  a and a′ from, a + b + g = 180° and a′ + b′ + g = 180°, so,  a = 180° - (b + g) = 180° - 96°11′27″ = 83°48′33″
and   a = 180° - (b′ + g) = 180° - (180° - b + g) = b - g = 65° - 31°11′27″ = 33°48′33″. Solving the oblique triangle - use of the sine law and the cosine law    Geometry and use of trigonometry contents - A 