Question

solve for x.
43°
8
7
x°
law of sines: \\(\frac{\sin a}{a} = \frac{\sin b}{b} = \frac{\sin c}{c}\\)
round your answer to the nearest hundredth.
\\(x = ?°\\)

Explanation:

Step1: Identify sides and angles

Let the triangle have angle \( A = 43^\circ \), side \( a = 7 \) (opposite angle \( A \)), side \( b = 8 \) (opposite angle \( B \)), and angle \( x \) (opposite side \( 7 \)? Wait, no: Wait, the side with length 8 is opposite to angle \( x \)? Wait, no, let's label correctly. Let's say: angle at the top is \( 43^\circ \), the side opposite to it is \( 7 \) (the base), the side with length 8 is one of the other sides, and the angle \( x \) is at the bottom right, opposite to the side of length 8? Wait, no, the triangle has sides: one side is 8 (left), base is 7 (bottom), and the other side (right) is opposite the \( 43^\circ \) angle? Wait, no, let's use Law of Sines correctly. Let's denote:

Let angle \( A = 43^\circ \), side \( a = 7 \) (opposite angle \( A \)), side \( b = 8 \) (opposite angle \( B \)), and angle \( C = x \), side \( c \) (but we don't need that). Wait, no, the side with length 8 is adjacent to the \( 43^\circ \) angle? Wait, the triangle: vertex at top: \( 43^\circ \), left side: 8, bottom side: 7, right side: let's call it \( c \), and angle at bottom right is \( x \). So, side opposite \( 43^\circ \) is 7 (bottom side), side opposite \( x \) is 8 (left side)? Wait, no, left side is 8, bottom is 7, so angle at bottom left: let's see, the angles are: top: \( 43^\circ \), bottom left: let's say \( y \), bottom right: \( x \). Then sides: opposite \( 43^\circ \) is 7 (bottom side), opposite \( x \) is 8 (left side), opposite \( y \) is the right side. So using Law of Sines: \( \frac{\sin(43^\circ)}{7} = \frac{\sin(x)}{8} \).

Step2: Apply Law of Sines

So, \( \frac{\sin(43^\circ)}{7} = \frac{\sin(x)}{8} \)

Multiply both sides by 8: \( \sin(x) = \frac{8 \times \sin(43^\circ)}{7} \)

Calculate \( \sin(43^\circ) \approx 0.681998 \)

So, \( \sin(x) = \frac{8 \times 0.681998}{7} \approx \frac{5.455984}{7} \approx 0.779426 \)

Then, \( x = \arcsin(0.779426) \approx 51.23^\circ \) (Wait, but we need to check if there's another possible angle, but since the side 8 is longer than 7, the angle opposite (x) should be larger than 43°, but let's check. Wait, maybe I labeled the sides wrong. Wait, maybe the side with length 8 is opposite the \( 43^\circ \) angle? No, the base is 7, so opposite \( 43^\circ \) is 7. Wait, let's re-express:

Let me define:

- Angle \( A = 43^\circ \)
- Side \( a = 7 \) (opposite angle \( A \))
- Side \( b = 8 \) (opposite angle \( B \), which is \( x \))

Then Law of Sines: \( \frac{\sin(A)}{a} = \frac{\sin(B)}{b} \)

So \( \frac{\sin(43^\circ)}{7} = \frac{\sin(x)}{8} \)

So \( \sin(x) = \frac{8 \times \sin(43^\circ)}{7} \)

Calculate \( \sin(43^\circ) \approx 0.681998 \)

So \( 8 \times 0.681998 = 5.455984 \)

Divide by 7: \( 5.455984 / 7 ≈ 0.779426 \)

Then \( x = \arcsin(0.779426) ≈ 51.23^\circ \) (using calculator: \( \arcsin(0.7794) \approx 51.23 \))

Wait, but let's check if the triangle is valid. The sum of angles should be 180. So angle \( A = 43^\circ \), angle \( x ≈ 51.23^\circ \), so the third angle is \( 180 - 43 - 51.23 = 85.77^\circ \), and the side opposite that angle would be \( \frac{7 \times \sin(85.77^\circ)}{\sin(43^\circ)} ≈ \frac{7 \times 0.997}{0.682} ≈ \frac{6.979}{0.682} ≈ 10.23 \), which is longer than 8, but that's okay. Alternatively, maybe I mixed up the sides. Wait, maybe the side with length 8 is opposite the \( 43^\circ \) angle? Then \( a = 8 \), angle \( A = 43^\circ \), side \( b = 7 \), angle \( B = x \). Then Law of Sines: \( \frac{\sin(43^\circ)}{8} = \frac{\sin(x)}{7} \). Then \( \sin(x) = \frac{7 \times \sin(43^\circ)}{8} ≈ \frac{7 \times 0.682}{8} ≈ \frac{4.774}{8} ≈ 0.59675 \), so \( x = \arcsin(0.59675) ≈ 36.67^\circ \). Wait, that's different. So which is correct?

Wait, the diagram: the left side is 8, bottom is 7, top angle is 43°. So the side opposite the top angle (43°) is the bottom side (7). So side \( a = 7 \) (opposite 43°), side \( b = 8 \) (opposite angle \( x \), which is at the bottom right). So then Law of Sines: \( \frac{\sin(43°)}{7} = \frac{\sin(x)}{8} \). So that's the first case. So \( \sin(x) = (8/7) \sin(43°) ≈ (8/7)*0.682 ≈ 0.779 \), so \( x ≈ 51.23° \). Let's check with calculator:

Calculate \( \sin(43°) ≈ 0.6819983601 \)

\( 8 * 0.6819983601 = 5.455986881 \)

\( 5.455986881 / 7 ≈ 0.7794266973 \)

\( \arcsin(0.7794266973) ≈ 51.23° \) (using calculator: arcsin(0.7794) is approximately 51.23 degrees)

So that's the calculation.

Answer:

\( \boxed{51.23} \) (rounded to nearest hundredth)