Question

solving for side lengths of right triangles
identifying trigonometric ratios to estimate side lengths
what are the correct trigonometric ratios that could be used to determine the length of ln? choose two correct answers.
cos (70°) = ln / 8
tan (70°) = ln / mn
sin (20°) = 8 / ln

Explanation:

Step1: Recall trigonometric ratio definitions

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Analyze $\cos(70^{\circ})$

If one of the non - right angles is $70^{\circ}$, and assuming the hypotenuse is $8$ and $LN$ is the adjacent side to the $70^{\circ}$ angle, then $\cos(70^{\circ})=\frac{LN}{8}$ is correct according to the cosine ratio definition.

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

If $LN$ is the opposite side and $MN$ is the adjacent side to the $70^{\circ}$ angle, then $\tan(70^{\circ})=\frac{LN}{MN}$ is correct according to the tangent ratio definition.

Step4: Analyze $\sin(20^{\circ})$

If the other non - right angle is $20^{\circ}$, $\sin(20^{\circ})=\frac{LN}{8}$ would be correct if $LN$ is the opposite side and $8$ is the hypotenuse. But the given expression $\sin(20^{\circ})=\frac{8}{LN}$ is incorrect.

Answer:

$\cos(70^{\circ})=\frac{LN}{8}$, $\tan(70^{\circ})=\frac{LN}{MN}$