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}\\)
image of a right triangle lnm with right angle at n, angle at l is 70°, angle at m is 20°, and hypotenuse lm is 8

Explanation:

Step1: Label sides for ∠L=70°

In right triangle LNM:

• Hypotenuse (opposite right angle): $LM = 8$
• Adjacent to ∠70°: $LN$
• Opposite to ∠70°: $MN$

Trig ratios:
$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, so $\cos(70^\circ) = \frac{LN}{8}$

Step2: Label sides for ∠M=20°

• Hypotenuse: $LM = 8$
• Opposite to ∠20°: $LN$

Trig ratios:
$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$, so $\sin(20^\circ) = \frac{LN}{8}$

Step3: Verify other options

• $\cos(70^\circ)=\frac{8}{LN}$: Incorrect (swaps adjacent/hypotenuse)
• $\tan(70^\circ)=\frac{LN}{MN}$: Incorrect (tan is opposite/adjacent, $\tan(70^\circ)=\frac{MN}{LN}$)
• $\sin(20^\circ)=\frac{8}{LN}$: Incorrect (swaps opposite/hypotenuse)

Answer:

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