Question

use the law of sines to find the value of w.
diagram of triangle uvw: angle at u is 31°, angle at w is 39°, side vw is 3.3 cm, side w is uv
law of sines: (\frac{sin(a)}{a} = \frac{sin(b)}{b} = \frac{sin(c)}{c})
what is the best approximation of the value of w?
options: 1.4 cm, 6.0 cm, 7.3 cm, 4.0 cm

Explanation:

Step1: Recall Law of Sines

The Law of Sines states that for a triangle with angles \( A, B, C \) and opposite sides \( a, b, c \) respectively, \(\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\). In triangle \( UVW \), let's identify the angles and sides. Angle at \( U \) is \( 31^\circ \), angle at \( W \) is \( 39^\circ \), and the side opposite angle \( W \) (i.e., side \( UV \))? Wait, no: side opposite angle \( U \) is \( VW = 3.3 \) cm, side opposite angle \( W \) is \( UV = w \)? Wait, no, let's label correctly. Let's denote: angle \( U = 31^\circ \), angle \( W = 39^\circ \), side \( VW = 3.3 \) cm (which is opposite angle \( U \)), and side \( UW = w \)? Wait, no, the side opposite angle \( W \) is \( UV \), and side opposite angle \( U \) is \( VW \). Wait, maybe better to assign: let \( \angle U = 31^\circ \), \( \angle W = 39^\circ \), side \( VW = 3.3 \) cm (opposite \( \angle U \)), and side \( UW = w \)? No, wait, the side we need to find is \( w \), which is opposite angle \( W \)? Wait, no, looking at the triangle: vertex \( U \), \( V \), \( W \). So side \( UV \) is opposite angle \( W \), side \( VW \) is opposite angle \( U \), and side \( UW \) is opposite angle \( V \). Wait, first, let's find angle \( V \). Since the sum of angles in a triangle is \( 180^\circ \), \( \angle V = 180^\circ - 31^\circ - 39^\circ = 110^\circ \). But maybe we don't need that. Wait, the Law of Sines: \(\frac{\sin(\angle U)}{VW}=\frac{\sin(\angle W)}{UV}\). Wait, \( \angle U = 31^\circ \), opposite side \( VW = 3.3 \) cm; \( \angle W = 39^\circ \), opposite side \( UV = w \). So according to Law of Sines: \(\frac{\sin(31^\circ)}{3.3}=\frac{\sin(39^\circ)}{w}\).

Step2: Solve for \( w \)

Cross-multiplying, we get \( w\times\sin(31^\circ)=3.3\times\sin(39^\circ) \). Then \( w = \frac{3.3\times\sin(39^\circ)}{\sin(31^\circ)} \). Now, calculate the sines: \( \sin(31^\circ)\approx0.5150 \), \( \sin(39^\circ)\approx0.6293 \). Plugging in: \( w=\frac{3.3\times0.6293}{0.5150} \). First, calculate numerator: \( 3.3\times0.6293\approx2.0767 \). Then divide by \( 0.5150 \): \( 2.0767\div0.5150\approx4.03 \), which is approximately \( 4.0 \) cm.

Answer:

4.0 cm