Questions: Given a=12 cm, b=19 cm, and γ=87°, find the angles of all possible triangles. Round answers to the nearest degree. c ≈ 7 cm, α ≈ 33°, β ≈ 60° c ≈ 22 cm, α ≈ 33°, β ≈ 60° c ≈ 7 cm, α ≈ 60°, β ≈ 33° c ≈ 22 cm, α ≈ 60°, β ≈ 33°

Solution Steps

We are given the following values for the triangle:

• \( a = 12 \, \text{cm} \)
• \( b = 19 \, \text{cm} \)
• \( \gamma = 87^\circ \)

Using the Law of Cosines, we can find the length of side \( c \): \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma) \] Substituting the values: \[ c^2 = 12^2 + 19^2 - 2 \cdot 12 \cdot 19 \cdot \cos(87^\circ) \] Calculating this gives us two possible lengths for \( c \): \[ c_1 \approx 22 \, \text{cm} \quad \text{and} \quad c_2 \approx 23 \, \text{cm} \]

Using the Law of Sines, we can find the angles \( \alpha \) and \( \beta \) for each case of \( c \).

For \( c_1 \): \[ \frac{a}{\sin(\alpha_1)} = \frac{c_1}{\sin(\gamma)} \] This leads to: \[ \sin(\alpha_1) = \frac{a \cdot \sin(\gamma)}{c_1} \] Calculating gives: \[ \alpha_1 \approx 33^\circ \] Then, using the fact that the sum of angles in a triangle is \( 180^\circ \): \[ \beta_1 = 180^\circ - \alpha_1 - \gamma \approx 60^\circ \]

For \( c_2 \): \[ \frac{a}{\sin(\alpha_2)} = \frac{c_2}{\sin(\gamma)} \] This leads to: \[ \sin(\alpha_2) = \frac{a \cdot \sin(\gamma)}{c_2} \] Calculating gives: \[ \alpha_2 \approx 31^\circ \] Then, using the sum of angles: \[ \beta_2 = 180^\circ - \alpha_2 - \gamma \approx 56^\circ \]

The possible triangles yield the following results:

1. For \( c_1 \approx 22 \, \text{cm} \):

   • \( \alpha_1 \approx 33^\circ \)
   • \( \beta_1 \approx 60^\circ \)

2. For \( c_2 \approx 23 \, \text{cm} \):

   • \( \alpha_2 \approx 31^\circ \)
   • \( \beta_2 \approx 56^\circ \)

Final Answer