Answer:
[tex]AB \approx 4.054[/tex]
Step-by-step explanation:
The trigonometric equations associated with the figure are, respectively:
[tex]AB = 4\cdot \cos 73^{\circ} + 3\cdot \cos \alpha[/tex]
[tex]4\cdot \sin 73^{\circ} = 3 +3\cdot \sin \alpha[/tex]
The components with the unknown angle are cleared and used in the fundamental trigonometric relation:
[tex]3\cdot \cos \alpha = AB - 4\cdot \cos 73^{\circ}[/tex]
[tex]3\cdot \sin \alpha = 4\cdot \sin 73^{\circ} - 3[/tex]
[tex]9\cdot \cos^{2}\alpha + 9\cdot \sin^{2}\alpha =(AB-4\cdot \cos 73^{\circ})^{2}+(4\cdot \sin 73^{\circ}-3)^{2}[/tex]
[tex]9 = AB^{2}-2.339\cdot AB + 1.368 + 0.681[/tex]
The following second-order polynomial is presented below:
[tex]AB^{2}-2.339\cdot AB -6.951 = 0[/tex]
Roots of the polynomial are described hereafter:
[tex]AB_{1} \approx 4.054[/tex] and [tex]AB_{2} \approx -1.715[/tex]
Only the first root is reasonable, as length is a positive variable. The length is [tex]AB \approx 4.054[/tex].
