Questions: Find the length of side a.

Transcript text: Find the length of side a.

Solution

Solution Steps

Step 1: Identify the Given Information

We are given a triangle with:

Angle A = 60°
Angle B = 45°
Side opposite to angle B (b) = 6

Step 2: Use the Law of Sines

The Law of Sines states: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] We need to find side \(a\). We know: \[ \frac{a}{\sin(60°)} = \frac{6}{\sin(45°)} \]

Step 3: Calculate the Sines of the Angles

\[ \sin(60°) = \frac{\sqrt{3}}{2} \] \[ \sin(45°) = \frac{\sqrt{2}}{2} \]

Step 4: Set Up the Equation

\[ \frac{a}{\frac{\sqrt{3}}{2}} = \frac{6}{\frac{\sqrt{2}}{2}} \]

Step 5: Solve for \(a\)

\[ a = 6 \cdot \frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{2}}{2}} = 6 \cdot \frac{\sqrt{3}}{\sqrt{2}} = 6 \cdot \frac{\sqrt{6}}{2} = 3\sqrt{6} \]

Step 6: Approximate the Value

\[ 3\sqrt{6} \approx 3 \cdot 2.45 = 7.35 \]

Final Answer

The length of side \(a\) is approximately 7.35, but this does not match any of the given options. Rechecking the steps, it seems there might be a mistake in the approximation or the given options might be incorrect. The closest option to our calculated value is not present.

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