Answer:
a) ∠ABC = 54°
a) ∠ABC = 43°
Step-by-step explanation:
Trigonometric ratios
[tex]\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}[/tex]
where:
- [tex]\theta[/tex] is the angle
- O is the side opposite the angle
- A is the side adjacent the angle
- H is the hypotenuse (the side opposite the right angle)
Part (a)
Use the cos trig ratio to find an expression for the measure of BD:
[tex]\implies \sf \cos (38^{\circ})=\dfrac{BD}{24.3}[/tex]
[tex]\implies \sf BD=24.3\cos (38^{\circ})[/tex]
Use the cos trig ratio and the found length of BD to find angle ABD:
[tex]\sf \implies \cos(ABD)=\dfrac{BD}{19.9}[/tex]
[tex]\sf \implies \angle ABD=\cos^{-1}\left(\dfrac{24.3\cos (38^{\circ})}{19.9}\right)[/tex]
[tex]\sf \implies \angle ABD=15.79446612...^{\circ}[/tex]
Therefore:
[tex]\sf \implies \angle ABC=\angle ADB + 38^{\circ}[/tex]
[tex]\sf \implies \angle ABC=15.79446612...^{\circ}+ 38^{\circ}[/tex]
[tex]\sf \implies \angle ABC=54^{\circ}\:\:(nearest\:degree)[/tex]
Part (b)
Use the tan trig ratio to find angle CAD:
[tex]\implies \sf \tan(CAD)=\dfrac{4.9}{7.4}[/tex]
[tex]\implies \sf \angle CAD=\tan^{-1} \left(\dfrac{4.9}{7.4}\right)[/tex]
[tex]\implies \sf \angle CAD=33.5110188...^{\circ}[/tex]
Therefore:
[tex]\sf \implies \angle BAD=13^{\circ}+33.5110188...^{\circ}[/tex]
[tex]\sf \implies \angle BAD=46.5110188...^{\circ}[/tex]
∠ABC = ∠ABD
Interior angles of a triangle sum to 180°
[tex]\sf \implies \angle ABD + \angle BAD + \angle BDA=180^{\circ}[/tex]
[tex]\sf \implies \angle ABC + 46.5110188...^{\circ} + 90^{\circ}=180^{\circ}[/tex]
[tex]\sf \implies \angle ABC=43^{\circ}\:\:(nearest\:degree)[/tex]
