Question

identifying trigonometric ratios to estimate side lengths
what are the correct trigonometric ratios that could be used to determine the length of ln? check all that apply.
$sin(20^{circ})=\frac{ln}{8}$
$cos(70^{circ})=\frac{8}{ln}$
$\tan(70^{circ})=\frac{ln}{mn}$
$sin(20^{circ})=\frac{8}{ln}$
$cos(70^{circ})=\frac{ln}{8}$

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 $LMN$ with right - angle at $N$, $\angle L = 70^{\circ}$ and $\angle M=20^{\circ}$, and hypotenuse $LM = 8$.

Step2: Analyze $\sin(20^{\circ})$

For $\angle M = 20^{\circ}$, the opposite side to $\angle M$ is $LN$ and the hypotenuse is $LM = 8$. So, $\sin(20^{\circ})=\frac{LN}{8}$, which is correct.

Step3: Analyze $\cos(70^{\circ})$

For $\angle L = 70^{\circ}$, the adjacent side to $\angle L$ is $LN$ and the hypotenuse is $LM = 8$. So, $\cos(70^{\circ})=\frac{LN}{8}$, which is correct.

Step4: Analyze $\tan(70^{\circ})$

For $\angle L = 70^{\circ}$, $\tan(70^{\circ})=\frac{MN}{LN}$, not $\frac{LN}{MN}$, so this is incorrect.

Step5: Analyze $\sin(20^{\circ})=\frac{8}{LN}$

Since $\sin(20^{\circ})=\frac{LN}{8}$, not $\frac{8}{LN}$, this is incorrect.

Step6: Analyze $\cos(70^{\circ})=\frac{8}{LN}$

Since $\cos(70^{\circ})=\frac{LN}{8}$, not $\frac{8}{LN}$, this is incorrect.

Answer:

$\sin(20^{\circ})=\frac{LN}{8}$, $\cos(70^{\circ})=\frac{LN}{8}$