Question

law of sines: \\(\frac{\sin(a)}{a} = \frac{\sin(b)}{b} = \frac{\sin(c)}{c}\\)

in \\(\triangle fgh\\), \\(h = 10\\), \\(m\angle f = 65^\circ\\), and \\(m\angle g = 35^\circ\\). what is the length of \\(g\\)? use the law of sines to find the answer.

• 5.8 units
• 6.7 units
• 9.2 units
• 9.8 units

Explanation:

🆕 New Concept Discovered: Law of Sines
Using ratios to find missing sides or angles in non-right triangles

Step 1: Find the measure of angle H

The sum of angles in any triangle is always \( 180^\circ \). We can find the missing angle \( m\angle H \) by subtracting the known angles from \( 180^\circ \):

$$ m\angle H = 180^\circ - m\angle F - m\angle G $$

$$ m\angle H = 180^\circ - 65^\circ - 35^\circ = 80^\circ $$

Step 2: Set up the Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles:

$$ \frac{\sin(G)}{g} = \frac{\sin(H)}{h} $$

Substitute the known values into the formula:

$$ \frac{\sin(35^\circ)}{g} = \frac{\sin(80^\circ)}{10} $$

Step 3: Solve for g

Rearrange the equation to isolate \( g \):

$$ g = \frac{10 \cdot \sin(35^\circ)}{\sin(80^\circ)} $$

Calculate the trigonometric values:

• \( \sin(35^\circ) \approx 0.5736 \)
• \( \sin(80^\circ) \approx 0.9848 \)

Substitute these values back into the equation:

$$ g \approx \frac{10 \cdot 0.5736}{0.9848} \approx \frac{5.736}{0.9848} \approx 5.82 $$

Rounding to the nearest tenth gives \( 5.8 \) units.

Answer:

5.8 units