Question

which equations could be used to solve for the unknown lengths of △abc? choose two correct answers. this question requires at least 2 answers. (ac) sin(45°) = bc; sin(45°) = \\(\frac{9}{bc}\\); sin(45°) = \\(\frac{bc}{9}\\); 9 tan(45°) = ac; cos(45°) = \\(\frac{bc}{9}\\)

Explanation:

Step1: Identify triangle parts

In right $\triangle ABC$, $\angle C=90^\circ$, $\angle A=45^\circ$, $AB=9$ (hypotenuse), $BC$ = opposite to $\angle A$, $AC$ = adjacent to $\angle A$.

Step2: Apply sine definition

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
$\sin(45^\circ) = \frac{BC}{AB} = \frac{BC}{9}$

Step3: Rearrange sine for BC

From $\sin(45^\circ) = \frac{BC}{AC}$ (alternate: $\sin(45^\circ) = \frac{BC}{AC}$), rearrange to $AC\sin(45^\circ)=BC$

Step4: Eliminate incorrect options

• $\sin(45^\circ)=\frac{9}{BC}$: Swaps hypotenuse/opposite, wrong.
• $9\tan(45^\circ)=AC$: $\tan(45^\circ)=\frac{BC}{AC}$, so $AC=\frac{BC}{\tan(45^\circ)}$, wrong.
• $\cos(45^\circ)=\frac{BC}{9}$: $\cos(45^\circ)=\frac{AC}{9}$, wrong.

Answer:

1. $\sin(45^\circ) = \frac{BC}{9}$
2. $(AC)\sin(45^\circ) = BC$