Solve the triangle with \( c=15.65, a=13.99 \), and \( \alpha=49^{\circ} \) \( \gamma=\square \) degrees \( \beta=\square \) \( b=\square \) Round all answees to 2 places after the decimal point. Give angles in degrees (but do not type "degrees" in the answer box). As in the text, side \( a \) is opposite from angle \( \alpha \), side \( b \) is opposite from angle \( \beta \), and side \( c \) is opposite from angle \( \gamma \). If no such triangle exists, then enter DNE in each answer box. If multiple triangles exist, then enter the values of the missing sides and angles for each triangle, separated by commas.

To solve the triangle with the given values \( c = 15.65 \), \( a = 13.99 \), and \( \alpha = 49^\circ \), we can use the Law of Sines and the properties of triangles. ### Step 1: Use the Law of Sines The Law of Sines states that: \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \] We can first find angle \( \gamma \) using the Law of Sines: \[ \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \] Rearranging gives: \[ \sin(\gamma) = \frac{c \cdot \sin(\alpha)}{a} \] ### Step 2: Calculate \( \sin(\gamma) \) Substituting the known values: \[ \sin(\gamma) = \frac{15.65 \cdot \sin(49^\circ)}{13.99} \] Now, we will calculate \( \sin(49^\circ) \) and then \( \sin(\gamma) \). ### Step 3: Calculate \( \sin(49^\circ) \) Using a calculator: \[ \sin(49^\circ) \approx 0.7547 \] Now substituting this value into the equation for \( \sin(\gamma) \): \[ \sin(\gamma) = \frac{15.65 \cdot 0.7547}{13.99} \] ### Step 4: Calculate \( \sin(\gamma) \) Now we will perform the calculation: \[ \sin(\gamma) \approx \frac{11.817355}{13.99} \approx 0.8455 \] ### Step 5: Find \( \gamma \) Now we find \( \gamma \): \[ \gamma = \arcsin(0.8455) \] Calculating \( \gamma \): \[ \gamma \approx 57.73^\circ \] ### Step 6: Find \( \beta \) Now we can find \( \beta \) using the fact that the sum of angles in a triangle is \( 180^\circ \): \[ \beta = 180^\circ - \alpha - \gamma \] Substituting the known values: \[ \beta = 180^\circ - 49^\circ - 57.73^\circ \] Calculating \( \beta \): \[ \beta \approx 73.27^\circ \] ### Step 7: Find side \( b \) Now we can find side \( b \) using the Law of Sines again: \[ \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \] Rearranging gives: \[ b = \frac{c \cdot \sin(\beta)}{\sin(\gamma)} \] Substituting the known values: \[ b = \frac{15.65 \cdot \sin(73.27^\circ)}{0.8455} \] Calculating \( \sin(73.27^\circ) \): \[ \sin(73.27^\circ) \approx 0.9563 \] Now substituting this value into the equation for \( b \): \[ b \approx \frac{15.65 \cdot 0.9563}{0.8455} \] Calculating \( b \): \[ b \approx \frac{14.973195}{0.8455} \approx 17.71 \] ### Final Answers Now we can summarize the results: - \( \gamma \approx 57.73 \) - \( \beta \approx 73.27 \) - \( b \approx 17.71 \) Thus, the final answers rounded to two decimal places are: \[ \gamma = 57.73, \beta = 73.27, b = 17.71 \]