QuestionJuly 14, 2025

Solve the triangle ABC. a=2.2 b=3.5 c=5.4 Find the unknown angle C, the angle opposite side C. C=141.8 (Round to the nearest tenth as needed.) Find the unknown angle B, the angle opposite side b. B=square (Round to the nearest tenth as needed.)

Solve the triangle ABC. a=2.2 b=3.5 c=5.4 Find the unknown angle C, the angle opposite side C. C=141.8 (Round to the nearest tenth as needed.) Find the unknown angle B, the angle opposite side b. B=square (Round to the nearest tenth as needed.)
Solve the triangle ABC.
a=2.2 b=3.5 c=5.4
Find the unknown angle C, the angle opposite side C.
C=141.8 (Round to the nearest tenth as needed.)
Find the unknown angle B, the angle opposite side b.
B=square  (Round to the nearest tenth as needed.)

Solution
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Answer

B = 24.1^\circ Explanation 1. Use the Law of Cosines to find angle C **Formula:** c^2 = a^2 + b^2 - 2ab \cdot \cos(C) Substitute values: 5.4^2 = 2.2^2 + 3.5^2 - 2 \cdot 2.2 \cdot 3.5 \cdot \cos(C) Solve for \cos(C): 29.16 = 4.84 + 12.25 - 15.4 \cdot \cos(C) 29.16 = 17.09 - 15.4 \cdot \cos(C) 12.07 = -15.4 \cdot \cos(C) \cos(C) = -0.7844 Find C: C = \cos^{-1}(-0.7844) \approx 141.8^\circ 2. Use the Law of Sines to find angle B **Formula:** \frac{b}{\sin(B)} = \frac{c}{\sin(C)} Substitute known values: \frac{3.5}{\sin(B)} = \frac{5.4}{\sin(141.8^\circ)} Solve for \sin(B): \sin(B) = \frac{3.5 \cdot \sin(141.8^\circ)}{5.4} \sin(B) \approx \frac{3.5 \cdot 0.6293}{5.4} \sin(B) \approx 0.4077 Find B: B = \sin^{-1}(0.4077) \approx 24.1^\circ

Explanation

1. Use the Law of Cosines to find angle C<br /> **Formula:** $c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$ <br /> Substitute values: $5.4^2 = 2.2^2 + 3.5^2 - 2 \cdot 2.2 \cdot 3.5 \cdot \cos(C)$ <br /> Solve for $\cos(C)$: <br />$29.16 = 4.84 + 12.25 - 15.4 \cdot \cos(C)$ <br />$29.16 = 17.09 - 15.4 \cdot \cos(C)$ <br />$12.07 = -15.4 \cdot \cos(C)$ <br />$\cos(C) = -0.7844$ <br /> Find $C$: <br />$C = \cos^{-1}(-0.7844) \approx 141.8^\circ$<br /><br />2. Use the Law of Sines to find angle B<br /> **Formula:** $\frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ <br /> Substitute known values: $\frac{3.5}{\sin(B)} = \frac{5.4}{\sin(141.8^\circ)}$ <br /> Solve for $\sin(B)$: <br />$\sin(B) = \frac{3.5 \cdot \sin(141.8^\circ)}{5.4}$ <br />$\sin(B) \approx \frac{3.5 \cdot 0.6293}{5.4}$ <br />$\sin(B) \approx 0.4077$ <br /> Find $B$: <br />$B = \sin^{-1}(0.4077) \approx 24.1^\circ$
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