The sine rule of trigonometry helps us to equate the side of the triangles to the angles of the triangles. The given triangle can be solved as shown below.
The sine rule of trigonometry helps us to equate the side of the triangles to the angles of the triangles. It is given by the formula,
[tex]\dfrac{Sin\ A}{\alpha} =\dfrac{Sin\ B}{\beta} =\dfrac{Sin\ C}{\gamma}[/tex]
where Sin A is the angle and α is the length of the side of the triangle opposite to angle A,
Sin B is the angle and β is the length of the side of the triangle opposite to angle B,
Sin C is the angle and γ is the length of the side of the triangle opposite to angle C.
For the given triangle, using the sine rule the ratio of the angle and the sides of the triangle can be written as,
[tex]\dfrac{Sin\ A}{18} =\dfrac{Sin\ B}{11} =\dfrac{Sin\ C}{c}\\\\\dfrac{Sin\ 72^o}{18} =\dfrac{Sin\ y^o}{11} =\dfrac{Sin\ x^o}{c}[/tex]
Taking the first two ratios,
[tex]\dfrac{Sin\ 72^o}{18} =\dfrac{Sin\ y^o}{11}\\\\y = 35.54^o[/tex]
The sum of all the angles of a triangle is 180°.
72° + 35.54° + x° = 180°
x = 72.46°
Now, using the sine ratio,
[tex]\dfrac{Sin\ 72^o}{18} =\dfrac{Sin\ 72.46^o}{c}\\\\c = 18.046[/tex]
Hence, the given triangle is solved.
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