Question

writing equations using trigonometric ratios
which equations could be used to solve for the unknown lengths of △abc? check all that apply.
□ $\sin(45^{\circ}) = \frac{bc}{9}$
□ $\sin(45^{\circ}) = \frac{9}{bc}$
□ $9 \tan(45^{\circ}) = ac$
□ $(ac)\sin(45^{\circ}) = bc$
□ $\cos(45^{\circ}) = \frac{bc}{9}$
(image of right triangle abc with right angle at c, angle at a is 45°, hypotenuse ab = 9)

Explanation:

Step1: Identify sides for ∠A

In right △ABC, ∠C=90°, ∠A=45°, hypotenuse AB=9.
Opposite to ∠A: BC; Adjacent to ∠A: AC.

Step2: Test sine ratio for ∠A

Sine is opposite/hypotenuse.
$\sin(45^\circ)=\frac{BC}{AB}=\frac{BC}{9}$

Step3: Test tangent ratio for ∠A

Tangent is opposite/adjacent.
$\tan(45^\circ)=\frac{BC}{AC} \implies AC\tan(45^\circ)=BC$, so $9\tan(45^\circ)=AC$ is invalid.

Step4: Test rearranged sine ratio

From $\sin(45^\circ)=\frac{BC}{AC}$, we get $(AC)\sin(45^\circ)=BC$.

Step5: Test cosine ratio for ∠A

Cosine is adjacent/hypotenuse.
$\cos(45^\circ)=\frac{AC}{9}$, so $\cos(45^\circ)=\frac{BC}{9}$ is invalid.
$\sin(45^\circ)=\frac{9}{BC}$ is the reciprocal of the correct sine ratio, so invalid.

Answer:

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