To use the Law of Sines to solve for side \(a\) in triangle \(ABC\), we can start with the information given:
- \( \angle C = 35^\circ \)
- \( c = 42 \) yd
From the Law of Sines, we know:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
Rearranging this gives:
\[ a = c \cdot \frac{\sin A}{\sin C} \]
To proceed, we need to find angle \(A\) and then apply the Law of Sines accordingly.
Step 1: Calculate \(B\)
Since it's a right triangle, we know:
\[ A + B + C = 180^\circ \]
Given that \( \angle B = 90^\circ \):
\[ A + 90^\circ + 35^\circ = 180^\circ \]
So:
\[ A = 180^\circ - 90^\circ - 35^\circ = 55^\circ \]
Step 2: Calculate \(a\)
Now we have:
- \(A = 55^\circ\)
- \(C = 35^\circ\)
- \(c = 42\) yd
Plugging these values into the Law of Sines equation, we have:
\[ a = 42 \cdot \frac{\sin(55^\circ)}{\sin(35^\circ)} \]
Step 3: Calculate the sines
Using a calculator (make sure it’s in degrees):
- \( \sin(55^\circ) \approx 0.8192\)
- \( \sin(35^\circ) \approx 0.5736\)
Step 4: Plug in the values
Now substituting these in:
\[ a = 42 \cdot \frac{0.8192}{0.5736} \approx 42 \cdot 1.4281 \approx 60.0 \]
Step 5: Round and finalize
Therefore, rounding to the nearest yard, we obtain:
\[ a \approx 60 \text{ yd} \]
Thus, the length of side \(a\) is:
60 yd.
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