Question

solve the triangle.
a = 4.18 m c = 5.93 m b = 27.7°
what is the length of side b?
□ m
(simplify your answer. type an integer or a decimal. round to four decimal places if needed.)
what is the measure of angle a?
□°
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)
what is the measure of angle c?
□°
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)

Explanation:

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 \( a = 4.18 \, \text{m} \), \( c = 5.93 \, \text{m} \), and \( B = 27.7^\circ \), we substitute these values into the formula:

\( b^2 = (4.18)^2 + (5.93)^2 - 2 \times 4.18 \times 5.93 \times \cos(27.7^\circ) \)

First, calculate the squares and the cosine term:

\( (4.18)^2 = 17.4724 \)

\( (5.93)^2 = 35.1649 \)

\( \cos(27.7^\circ) \approx 0.8854 \) (using a calculator)

Now, calculate the product term:

\( 2 \times 4.18 \times 5.93 = 2 \times 24.8874 = 49.7748 \)

\( 49.7748 \times 0.8854 \approx 49.7748 \times 0.8854 \approx 44.07 \) (approximate value)

Now, substitute back into the formula for \( b^2 \):

\( b^2 = 17.4724 + 35.1649 - 44.07 \)

\( b^2 = 52.6373 - 44.07 = 8.5673 \)

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

\( b = \sqrt{8.5673} \approx 2.9270 \) (rounded to four decimal places)

Step 1: Apply the Law of Sines

The Law of Sines states that \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \). We know \( a = 4.18 \, \text{m} \), \( b \approx 2.9270 \, \text{m} \), and \( B = 27.7^\circ \).

Rearranging for \( \sin(A) \):

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

Substitute the known values:

\( \sin(A) = \frac{4.18 \times \sin(27.7^\circ)}{2.9270} \)

Calculate \( \sin(27.7^\circ) \approx 0.4659 \) (using a calculator)

\( 4.18 \times 0.4659 \approx 4.18 \times 0.4659 \approx 1.947 \)

Now, divide by \( b \):

\( \sin(A) = \frac{1.947}{2.9270} \approx 0.6652 \)

Step 2: Find angle \( A \)

Take the inverse sine (arcsin) of \( 0.6652 \):

\( A = \arcsin(0.6652) \approx 41.7^\circ \) (rounded to the nearest tenth)

Step 1: Use the fact that the sum of angles in a triangle is \( 180^\circ \)

In any triangle, \( A + B + C = 180^\circ \). We know \( A \approx 41.7^\circ \) and \( B = 27.7^\circ \).

So, \( C = 180^\circ - A - B \)

Substitute the known values:

\( C = 180^\circ - 41.7^\circ - 27.7^\circ \)

\( C = 180^\circ - 69.4^\circ = 110.6^\circ \) (rounded to the nearest tenth)

Answer:

\( 2.9270 \)

Measure of angle \( A \)