Question

solve the triangle with the given parts.
a = 98.9°, b = 9, c = 7
what is the length of side a?
(round to four decimal places.)
what is the measure of angle b?
(round to the nearest tenth.)
what is the measure of angle c?
(round to the nearest tenth.)

Explanation:

Step1: Find side \( a \) using the Law of Cosines

The Law of Cosines states that \( a^2 = b^2 + c^2 - 2bc \cos A \). Given \( A = 98.9^\circ \), \( b = 9 \), and \( c = 7 \), we substitute these values into the formula.
First, calculate \( b^2 = 9^2 = 81 \), \( c^2 = 7^2 = 49 \), and \( 2bc = 2 \times 9 \times 7 = 126 \). Then, \( \cos A = \cos(98.9^\circ) \approx -0.1545 \).
So, \( a^2 = 81 + 49 - 126 \times (-0.1545) = 130 + 19.467 = 149.467 \). Taking the square root, \( a = \sqrt{149.467} \approx 12.2257 \).

Step2: Find angle \( B \) using the Law of Sines

The Law of Sines is \( \frac{\sin B}{b} = \frac{\sin A}{a} \). We know \( b = 9 \), \( A = 98.9^\circ \), and \( a \approx 12.2257 \).
First, calculate \( \sin A = \sin(98.9^\circ) \approx 0.9890 \). Then, \( \sin B = \frac{b \sin A}{a} = \frac{9 \times 0.9890}{12.2257} \approx \frac{8.901}{12.2257} \approx 0.7281 \). Taking the inverse sine, \( B \approx \arcsin(0.7281) \approx 46.7^\circ \).

Step3: Find angle \( C \) using the triangle angle sum

The sum of angles in a triangle is \( 180^\circ \). So, \( C = 180^\circ - A - B \). Substituting \( A = 98.9^\circ \) and \( B \approx 46.7^\circ \), we get \( C = 180 - 98.9 - 46.7 = 34.4^\circ \).

Answer:

• Length of side \( a \): \( \approx 12.2257 \)
• Measure of angle \( B \): \( \approx 46.7^\circ \)
• Measure of angle \( C \): \( \approx 34.4^\circ \)