We are tasked with finding side $f$ in $\triangle DEF$, where the angles are:

$D = 71.5^\circ, \, E = 30.2^\circ, \, \text{and side } d = 7.4 \, \text{m}.$

Step 1: Find the third angle $F$

The sum of the interior angles in a triangle is $180^\circ$. Therefore: $F = 180^\circ - D - E$ $F = 180^\circ - 71.5^\circ - 30.2^\circ = 78.3^\circ$

Step 2: Apply the Law of Sines

The Law of Sines states: $\frac{d}{\sin D} = \frac{f}{\sin F}$ Substituting the known values: $\frac{7.4}{\sin 71.5^\circ} = \frac{f}{\sin 78.3^\circ}$

Step 3: Solve for $f$

Rearranging to solve for $f$: $f = \frac{7.4 \cdot \sin 78.3^\circ}{\sin 71.5^\circ}$

Using trigonometric values:

• $\sin 71.5^\circ \approx 0.946$
• $\sin 78.3^\circ \approx 0.981$

Substitute these values: $f = \frac{7.4 \cdot 0.981}{0.946} \approx \frac{7.2634}{0.946} \approx 7.68 \, \text{m}$

Final Answer:

$f \approx 7.68 \, \text{m}$

Would you like further clarification or additional help?

Questions to Explore:

1. How can the Law of Sines be applied to find other sides or angles?
2. What are other ways to solve this triangle if additional information is given?
3. How is the Law of Cosines different from the Law of Sines?
4. How do rounding errors affect trigonometric calculations?
5. Can you solve for side $e$ if required?

Tip:

Always double-check angle measurements in degrees or radians when performing tr