The length of AC is 16 km.
Solution:
Given data:
AB = c = 14 km, ∠A = 30° and ∠B = 89°
AC = b = ?
Let us first find angle C:
Sum of all angles in a triangle = 180°
m∠A+ m∠B + m∠C = 180°
30° + 89° + m∠C = 180°
119° + m∠C = 180°
Subtract 119° from both sides, we get
m∠C = 61°
To find the length of AC:
Using sine formula:
[tex]$\frac{b}{\sin B } =\frac{c}{\sin C}[/tex]
Substitute the given values in the formula.
[tex]$\frac{b}{\sin 89^\circ } =\frac{14}{\sin 61^\circ }[/tex]
Multiply by sin 89° on both sides.
[tex]$\sin 89^\circ \times \frac{b}{\sin 89^\circ } =\frac{14}{\sin 61^\circ } \times \sin 89^\circ[/tex]
[tex]$b=\frac{14}{0.8746 } \times0.9998[/tex]
[tex]b=16[/tex]
The length of AC is 16 km.
