Question

solve for c.
c
43°
a = 8
c = ?
55°
a
b
law of sines: (\frac{sin a}{a} = \frac{sin b}{b} = \frac{sin c}{c})
round your answer to the nearest hundredth.

Explanation:

Step1: Find angle B

In a triangle, the sum of angles is \(180^\circ\). So, \( \angle B = 180^\circ - 43^\circ - 55^\circ = 82^\circ \).

Step2: Apply Law of Sines

Law of Sines: \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \). Here, we use \( \frac{\sin A}{a} = \frac{\sin C}{c} \). Wait, no, correct: \( \frac{\sin B}{b} \)? Wait, no, \( a = 8 \) is opposite angle A? Wait, no, in triangle notation, side \( a \) is opposite angle \( A \), side \( b \) opposite angle \( B \), side \( c \) opposite angle \( C \). Wait, the triangle: angle at C is \( 43^\circ \), angle at A is \( 55^\circ \), side \( a = 8 \) (opposite angle A? Wait, no, side \( a \) is BC, opposite angle A. Wait, angle at A is \( 55^\circ \), angle at C is \( 43^\circ \), so angle at B is \( 180 - 55 - 43 = 82^\circ \). Side \( a = 8 \) (opposite angle A, so angle A is \( 55^\circ \), side \( a = 8 \); side \( c \) is opposite angle C (\( 43^\circ \)). So Law of Sines: \( \frac{a}{\sin A} = \frac{c}{\sin C} \). So \( c = \frac{a \cdot \sin C}{\sin A} \).
Substitute \( a = 8 \), \( \sin A = \sin 55^\circ \), \( \sin C = \sin 43^\circ \).
\( \sin 55^\circ \approx 0.8192 \), \( \sin 43^\circ \approx 0.6820 \).
So \( c = \frac{8 \cdot 0.6820}{0.8192} \approx \frac{5.456}{0.8192} \approx 6.66 \). Wait, wait, no: wait, angle A is \( 55^\circ \), side \( a = 8 \) (opposite angle A), angle C is \( 43^\circ \), side \( c \) opposite angle C. So formula is \( \frac{c}{\sin C} = \frac{a}{\sin A} \), so \( c = \frac{a \cdot \sin C}{\sin A} \).
Calculating: \( \sin 55^\circ \approx 0.819152 \), \( \sin 43^\circ \approx 0.681998 \).
\( c = \frac{8 \times 0.681998}{0.819152} \approx \frac{5.455984}{0.819152} \approx 6.66 \). Wait, but let's check again. Wait, angle B is \( 82^\circ \), side \( a = 8 \) (opposite angle A: \( 55^\circ \)), so \( \frac{a}{\sin A} = \frac{8}{\sin 55^\circ} \approx \frac{8}{0.8192} \approx 9.765 \). Then \( \frac{c}{\sin C} = 9.765 \), so \( c = 9.765 \times \sin 43^\circ \approx 9.765 \times 0.6820 \approx 6.66 \).

Answer:

\( \approx 6.66 \)