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°) = bc/9
□ sin(45°) = 9/bc
□ 9 tan(45°) = ac
□ (ac)sin(45°) = bc
□ cos(45°) = bc/9

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In right - triangle $ABC$ with $\angle A = 45^{\circ}$, the hypotenuse $AB = 9$.

Step2: Analyze $\sin(45^{\circ})$

$\sin(45^{\circ})=\frac{BC}{AB}$. Since $AB = 9$, we have $\sin(45^{\circ})=\frac{BC}{9}$.

Step3: Analyze $\tan(45^{\circ})$

$\tan(45^{\circ})=\frac{BC}{AC}$. Since $\tan(45^{\circ}) = 1$, we have $BC=AC$. Also, from $\sin(45^{\circ})=\frac{BC}{9}$, we know $BC = 9\sin(45^{\circ})$ and $AC = 9\sin(45^{\circ})$. And $9\tan(45^{\circ})=9\times1 = 9
eq AC$.

Step4: Analyze $\cos(45^{\circ})$

$\cos(45^{\circ})=\frac{AC}{AB}=\frac{AC}{9}$, not $\frac{BC}{9}$.

Answer:

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