Question

solving for side lengths of right triangles
writing equations using trigonometric ratios
which equations could be used to solve for the unknown lengths of △abc? choose two correct answers.
(ac)sin(45°)=bc
9 tan(45°)=ac
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 $\angle A = 45^{\circ}$ and hypotenuse $AB = 9$.

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

We know that $\sin A=\frac{BC}{AB}$. Since $A = 45^{\circ}$ and $AB = 9$, $\sin(45^{\circ})=\frac{BC}{9}$, which can be rewritten as $9\sin(45^{\circ})=BC$. Also, $\sin(45^{\circ})=\frac{BC}{AC}$ gives $(AC)\sin(45^{\circ})=BC$.

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

$\cos A=\frac{AC}{AB}$. Since $A = 45^{\circ}$ and $AB = 9$, $\cos(45^{\circ})=\frac{AC}{9}$, or $AC = 9\cos(45^{\circ})$. And $\cos(45^{\circ})=\frac{AC}{9}$, also $\cos(45^{\circ})=\frac{BC}{9}$ (because in a $45 - 45-90$ triangle $AC = BC$).

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

$\tan A=\frac{BC}{AC}$. Since $A = 45^{\circ}$, $\tan(45^{\circ}) = 1$. If we consider the relationships with the given side length, the correct equations for side - length relationships are based on the hypotenuse and the angle. The equation $9\tan(45^{\circ})=AC$ is incorrect as $\tan(45^{\circ})=\frac{BC}{AC}$ and using the hypotenuse relationship is more appropriate here.

Answer:

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