Question

Can someone please help me answer these 2 questions with a full explanation so I can do the rest on my own? will give brainliest to the best description of how you got your answer.

Answer 1

Answer:

a)  ∠ABC = 54°

a)  ∠ABC = 43°

Step-by-step explanation:

Trigonometric ratios

$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$

where:

• $\theta$ 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:

$\implies \sf \cos (38^{\circ})=\dfrac{BD}{24.3}$

$\implies \sf BD=24.3\cos (38^{\circ})$

Use the cos trig ratio and the found length of BD to find angle ABD:

$\sf \implies \cos(ABD)=\dfrac{BD}{19.9}$

$\sf \implies \angle ABD=\cos^{-1}\left(\dfrac{24.3\cos (38^{\circ})}{19.9}\right)$

$\sf \implies \angle ABD=15.79446612...^{\circ}$

Therefore:

$\sf \implies \angle ABC=\angle ADB + 38^{\circ}$

$\sf \implies \angle ABC=15.79446612...^{\circ}+ 38^{\circ}$

$\sf \implies \angle ABC=54^{\circ}\:\:(nearest\:degree)$

Part (b)

Use the tan trig ratio to find angle CAD:

$\implies \sf \tan(CAD)=\dfrac{4.9}{7.4}$

$\implies \sf \angle CAD=\tan^{-1} \left(\dfrac{4.9}{7.4}\right)$

$\implies \sf \angle CAD=33.5110188...^{\circ}$

Therefore:

$\sf \implies \angle BAD=13^{\circ}+33.5110188...^{\circ}$

$\sf \implies \angle BAD=46.5110188...^{\circ}$

∠ABC = ∠ABD

Interior angles of a triangle sum to 180°

$\sf \implies \angle ABD + \angle BAD + \angle BDA=180^{\circ}$

$\sf \implies \angle ABC + 46.5110188...^{\circ} + 90^{\circ}=180^{\circ}$

$\sf \implies \angle ABC=43^{\circ}\:\:(nearest\:degree)$