Question

writing equations using trigonometric ratios
which equations could be used to solve for the unknown lengths of $\triangle abc$? check all that apply.
$\square \\ \sin(45^{\circ}) = \frac{bc}{9}$
$\square \\ \sin(45^{\circ}) = \frac{9}{bc}$
$\square \\ 9\tan(45^{\circ}) = ac$
$\square \\ (ac)\sin(45^{\circ}) = bc$
$\square \\ \cos(45^{\circ}) = \frac{ac}{9}$

Explanation:

Step1: Define trigonometric ratios

For $\angle A = 45^\circ$ in right $\triangle ABC$:

• $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Opposite to $\angle A$: $BC$, Adjacent to $\angle A$: $AC$, Hypotenuse: $AB=9$

Step2: Verify $\sin(45^\circ)$ options

$\sin(45^\circ) = \frac{BC}{9}$ (matches first option). The second option $\sin(45^\circ)=\frac{9}{BC}$ is incorrect.

Step3: Verify $\tan(45^\circ)$ option

$\tan(45^\circ) = \frac{BC}{AC} \implies BC = AC \tan(45^\circ)$, or rearranged: $AC \tan(45^\circ)=BC$. The option $9\tan(45^\circ)=AC$ is incorrect.

Step4: Verify $(AC)\sin(45^\circ)=BC$

From $\sin(45^\circ)=\frac{BC}{9}$ and $\cos(45^\circ)=\frac{AC}{9} \implies AC=9\cos(45^\circ)$. Substitute: $9\cos(45^\circ)\sin(45^\circ)=BC$, which does not match. This option is incorrect.

Step5: Verify $\cos(45^\circ)$ option

$\cos(45^\circ) = \frac{AC}{9}$ (matches last option).

Answer:

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