Question

solve the triangle.
a = 8.437 in c = 6.816 in b = 78.77°
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 \( a = 8.437 \) in, \( c = 6.816 \) in, and \( B = 78.77^\circ \), we first calculate \( \cos(78.77^\circ) \).

\( \cos(78.77^\circ) \approx 0.193 \) (using a calculator)

Now, substitute the values into the formula:

\( b^2 = (8.437)^2 + (6.816)^2 - 2 \times 8.437 \times 6.816 \times 0.193 \)

Step 2: Calculate each term

• \( (8.437)^2 \approx 71.183 \)
• \( (6.816)^2 \approx 46.458 \)
• \( 2 \times 8.437 \times 6.816 \times 0.193 \approx 2 \times 8.437 \times 6.816 \times 0.193 \approx 21.944 \)

Now, sum the first two terms and subtract the third:

\( b^2 \approx 71.183 + 46.458 - 21.944 = 95.697 \)

Step 3: Take the square root

\( b = \sqrt{95.697} \approx 9.782 \) in

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

Step 1: Apply the Law of Sines

The Law of Sines states that \( \frac{\sin(A)}{a} = \frac{\sin(B)}{b} \).

We know \( a = 8.437 \) in, \( b \approx 9.782 \) in, and \( B = 78.77^\circ \).

First, find \( \sin(78.77^\circ) \approx 0.981 \) (using a calculator)

Then, solve for \( \sin(A) \):

\( \sin(A) = \frac{a \times \sin(B)}{b} = \frac{8.437 \times 0.981}{9.782} \)

Step 2: Calculate the numerator

\( 8.437 \times 0.981 \approx 8.277 \)

Step 3: Solve for \( \sin(A) \)

\( \sin(A) \approx \frac{8.277}{9.782} \approx 0.846 \)

Step 4: Find angle \( A \)

\( A = \arcsin(0.846) \approx 57.82^\circ \) (using a calculator)

Part 3: Find the measure of angle \( C \)

Step 1: Use the triangle angle sum

The sum of the angles in a triangle is \( 180^\circ \). So, \( A + B + C = 180^\circ \).

We know \( A \approx 57.82^\circ \) and \( B = 78.77^\circ \).

Step 2: Solve for \( C \)

\( C = 180^\circ - A - B = 180^\circ - 57.82^\circ - 78.77^\circ = 43.41^\circ \)

Answer:

s:

• Length of side \( b \): \( \boxed{9.782} \) in
• Measure of angle \( A \): \( \boxed{57.82}^\circ \)
• Measure of angle \( C \): \( \boxed{43.41}^\circ \)