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Chapter 5: Problem 33

Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=45^{\circ}, \quad a=b=1$$

Short Answer

Expert verified
The triangle has one possible solution, and the measures are \(A = B = 45^{\circ}\), \(C = 90^{\circ}\) and \(c \approx 1.41\).

Step by step solution

01

Finding angle B

Apply the Law of Sines. Since \(a = b\), it means that angle \(A\) and angle \(B\) are the same. So, \(B = 45^{\circ}\).
02

Finding angle C

The sum of angles in a triangle is \(180^{\circ}\). So, subtract the known angles from \(180^{\circ}\) to find the unknown angle. Therefore, \(C = 180^{\circ} - A - B = 180^{\circ} - 45^{\circ} - 45^{\circ} = 90^{\circ}\).
03

Calculating side c

Now, use the Law of Sines to find side \(c\). \(\frac{c}{\sin C} = \frac{a}{\sin A}\). Substituting the known values, we get \(c = a \times \frac{\sin C}{\sin A} = 1 \times \frac{sin 90^{\circ}}{sin 45^{\circ}}\). After calculation, one can find that \(c \approx 1.41\).

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