Question

solve the triangle.
a = 7.714 in c = 6.338 in b = 73.05°
what is the length of side b?
□ in
(round to the nearest thousandth as needed.)
what is the measure of angle a?
□°
(round to the nearest hundredth as needed.)
what is the measure of angle c?
□°
(round to the nearest hundredth as needed.)

Explanation:

Part 1: Find the length of side \( b \)

Step 1: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \( a \), \( b \), \( c \) and the angle \( B \) opposite side \( b \), \( b^2 = a^2 + c^2 - 2ac \cos(B) \).
Given \( a = 7.714 \) in, \( c = 6.338 \) in, and \( B = 73.05^\circ \), we substitute these values into the formula.
First, calculate \( a^2 \), \( c^2 \), and \( 2ac \cos(B) \):
\( a^2 = (7.714)^2 \approx 59.505796 \)
\( c^2 = (6.338)^2 \approx 40.170244 \)
\( 2ac = 2 \times 7.714 \times 6.338 \approx 98.344304 \)
\( \cos(73.05^\circ) \approx \cos(73.05) \approx 0.2907 \) (using a calculator)
Then, \( 2ac \cos(B) \approx 98.344304 \times 0.2907 \approx 28.600 \)
Now, \( b^2 = 59.505796 + 40.170244 - 28.600 \approx 71.076 \)

Step 2: Take the square root to find \( b \)

\( b = \sqrt{71.076} \approx 8.430 \) (rounded to the nearest thousandth)

Step 1: Apply the Law of Sines

The Law of Sines states that \( \frac{\sin(A)}{a} = \frac{\sin(B)}{b} \). We can solve for \( \sin(A) \):
\( \sin(A) = \frac{a \sin(B)}{b} \)
We know \( a = 7.714 \), \( B = 73.05^\circ \), and \( b \approx 8.430 \).
First, calculate \( \sin(73.05^\circ) \approx \sin(73.05) \approx 0.9563 \)
Then, \( \sin(A) = \frac{7.714 \times 0.9563}{8.430} \approx \frac{7.377}{8.430} \approx 0.8751 \)

Step 2: Find the angle \( A \)

Take the inverse sine (arcsin) of \( 0.8751 \):
\( A = \arcsin(0.8751) \approx 61.00^\circ \) (rounded to the nearest hundredth)

Step 1: Use the triangle angle sum property

The sum of the angles in a triangle is \( 180^\circ \). So, \( A + B + C = 180^\circ \).
We know \( A \approx 61.00^\circ \) and \( B = 73.05^\circ \).

Step 2: Solve for \( C \)

\( C = 180^\circ - A - B \)
\( C = 180 - 61.00 - 73.05 = 45.95^\circ \) (rounded to the nearest hundredth)

Answer:

\( 8.430 \)

Part 2: Find the measure of angle \( A \)