Questions: Using the Law of Sines to solve the all possible triangles if angle A=120 degrees, a=29, b=14. If no answer exists, enter DNE for all answers. angle B is degrees angle C is degrees c= Assume angle A is opposite side a, angle B is opposite side b, and angle C is opposite side c.

Solution Steps

To solve the triangle using the Law of Sines, we will:

1. Use the Law of Sines to find $\sin(B)$.
2. Determine if there are any possible values for $\angle B$.
3. Calculate $\angle C$ using the fact that the sum of angles in a triangle is $180^\circ$.
4. Use the Law of Sines again to find the length of side $c$.

We are given the following values for triangle \( ABC \):

• \( \angle A = 120^\circ \)
• \( a = 29 \)
• \( b = 14 \)

Using the Law of Sines, we find \( \sin(B) \) as follows: \[ \sin(B) = \frac{b \cdot \sin(A)}{a} = \frac{14 \cdot \sin(120^\circ)}{29} \] Calculating this gives: \[ \sin(B) \approx 0.4181 \]

Since \( \sin(B) \) is valid, we can find \( B \): \[ B \approx \arcsin(0.4181) \approx 24.71^\circ \] Additionally, the second possible angle is: \[ B' = 180^\circ - B \approx 155.29^\circ \]

Using the triangle angle sum property, we calculate \( C \) for both possible values of \( B \):

1. For \( B \approx 24.71^\circ \): \[ C \approx 180^\circ - A - B \approx 180^\circ - 120^\circ - 24.71^\circ \approx 35.29^\circ \]
2. For \( B' \approx 155.29^\circ \): \[ C' \approx 180^\circ - A - B' \approx 180^\circ - 120^\circ - 155.29^\circ \approx -95.29^\circ \] Since \( C' \) is negative, it is not a valid angle.

Using the Law of Sines again, we find \( c \) for the valid triangle: \[ c = \frac{a \cdot \sin(C)}{\sin(A)} = \frac{29 \cdot \sin(35.29^\circ)}{\sin(120^\circ)} \approx 19.34 \]

Final Answer