Math / Trigonometry

QuestionSolve the triangle. $\mathrm{a}=7.232 \text { in } \mathrm{c}=6.525 \text { in } B=73.27^{\circ}$
What is the length of side $b$ ? $\square$ in (Round to the nearest thousandth as needed.) What is the measure of angle A? $\square$ (Round to the nearest hundredth as needed.) What is the measure of angle C? $\square$ (Round to the nearest hundredth as needed.)

Studdy Solution

STEP 1

1. We are given a triangle with sides $a = 7.232$ inches, $c = 6.525$ inches, and angle $B = 73.27^\circ$ .
2. We need to find the length of side $b$ and the measures of angles $A$ and $C$ .

STEP 2

1. Use the Law of Cosines to find the length of side $b$ .
2. Use the Law of Sines to find the measure of angle $A$ .
3. Use the Triangle Angle Sum Theorem to find the measure of angle $C$ .

STEP 3

Use the Law of Cosines to find the length of side $b$ :
The Law of Cosines states:
$b^2 = a^2 + c^2 - 2ac \cdot \cos(B)$
Substitute the given values:
$b^2 = 7.232^2 + 6.525^2 - 2 \times 7.232 \times 6.525 \times \cos(73.27^\circ)$
Calculate $b$ :
$b^2 = 52.323424 + 42.590625 - 2 \times 7.232 \times 6.525 \times \cos(73.27^\circ)$
$b^2 = 94.914049 - 2 \times 7.232 \times 6.525 \times 0.29237$
$b^2 = 94.914049 - 27.563$
$b^2 = 67.351049$
$b = \sqrt{67.351049}$
$b \approx 8.204$

STEP 4

Use the Law of Sines to find the measure of angle $A$ :
The Law of Sines states:
$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$
Substitute the known values:
$\frac{7.232}{\sin(A)} = \frac{8.204}{\sin(73.27^\circ)}$
Solve for $\sin(A)$ :
$\sin(A) = \frac{7.232 \times \sin(73.27^\circ)}{8.204}$
Calculate $\sin(A)$ :
$\sin(A) = \frac{7.232 \times 0.9563}{8.204}$
$\sin(A) = \frac{6.914}{8.204}$
$\sin(A) \approx 0.8428$
Find $A$ :
$A \approx \sin^{-1}(0.8428)$
$A \approx 57.47^\circ$

STEP 5

Use the Triangle Angle Sum Theorem to find the measure of angle $C$ :
The Triangle Angle Sum Theorem states:
$A + B + C = 180^\circ$
Substitute the known values:
$57.47^\circ + 73.27^\circ + C = 180^\circ$
Solve for $C$ :
$C = 180^\circ - 57.47^\circ - 73.27^\circ$
$C \approx 49.26^\circ$
The length of side $b$ is:
$\boxed{8.204 \text{ in}}$
The measure of angle $A$ is:
$\boxed{57.47^\circ}$
The measure of angle $C$ is:
$\boxed{49.26^\circ}$

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