Questions: Solve the following triangle. B=45°, C=64°, b=39 inches A= (Simplify your answer.)

Transcript text: Solve the following triangle. \[ B=45^{\circ}, C=64^{\circ}, b=39 \text { inches } \] $A=$ $\square$ $\cdot$ (Simplify your answer.)

Solution

Solution Steps

To solve the triangle, we can use the fact that the sum of angles in a triangle is $180^\circ$. First, find angle $A$ by subtracting the given angles $B$ and $C$ from $180^\circ$. Then, use the Law of Sines to find the length of side $a$.

Step 1: Calculate Angle $ A $

To find angle $ A $, we use the fact that the sum of the angles in a triangle is $ 180^\circ $: \[ A = 180^\circ - B - C = 180^\circ - 45^\circ - 64^\circ = 71^\circ \]

Step 2: Apply the Law of Sines

Using the Law of Sines, we can find the length of side $ a $: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \] Substituting the known values: \[ \frac{a}{\sin(71^\circ)} = \frac{39}{\sin(45^\circ)} \]

Step 3: Solve for Side $ a $

Rearranging the equation to solve for $ a $: \[ a = b \cdot \frac{\sin(A)}{\sin(B)} = 39 \cdot \frac{\sin(71^\circ)}{\sin(45^\circ)} \approx 52.1494 \]

Final Answer

The values we found are: \[ A = 71^\circ \quad \text{and} \quad a \approx 52.1494 \] Thus, the final boxed answers are: \[ \boxed{A = 71^\circ} \] \[ \boxed{a \approx 52.1494} \]

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