Question

triangle efg is shown.

f = 32

e = 16

e

g = 20

f

g

use the triangle to complete the statements. for help, see this worked example.

the measure of ∠e is approximately °.

the ratio of sin(e) and e is approximately .

so the ratio of sin(g) and g is approximately .

the measure of ∠g is approximately °.

Explanation:

Step1: Identify sides for ∠E

Side opposite ∠E: \(e = 16\); sides adjacent/hypotenuse: \(f = 32\), \(g = 20\). Use Law of Cosines: \(e^2 = f^2 + g^2 - 2fg\cos E\)

Step2: Calculate cos E

Substitute values: \(16^2 = 32^2 + 20^2 - 2(32)(20)\cos E\) → \(256 = 1024 + 400 - 1280\cos E\) → \(256 = 1424 - 1280\cos E\) → \(1280\cos E = 1424 - 256 = 1168\) → \(\cos E = \frac{1168}{1280} = 0.9125\)

Step3: Find ∠E

\(∠E ≈ \arccos(0.9125) ≈ 24^\circ\)

Step4: Compute sin(E)/e

\(\sin(24^\circ) ≈ 0.4067\); ratio: \(\frac{0.4067}{16} ≈ 0.0254\)

Step5: Equal ratio for sin(G)/g

By Law of Sines: \(\frac{\sin E}{e} = \frac{\sin G}{g}\), so ≈0.0254

Step6: Find ∠G

\(\sin G ≈ 0.0254×20 ≈ 0.508\); \(∠G ≈ \arcsin(0.508) ≈ 31^\circ\)

Answer:

24
0.025
0.025
31