Questions: In triangle UVW, w=690 inches, m angle W=79° and m angle U=82°. Find the length of v, to the nearest 10th of an inch.

Solution Steps

To find the length of side \( v \) in triangle \( \triangle \mathrm{UVW} \), we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of the triangle. Given \( w = 690 \) inches, \( m \angle \mathrm{W} = 79^\circ \), and \( m \angle \mathrm{U} = 82^\circ \), we can find \( m \angle \mathrm{V} \) and then use the Law of Sines to find \( v \).

1. Calculate \( m \angle \mathrm{V} \) using the fact that the sum of angles in a triangle is \( 180^\circ \).
2. Apply the Law of Sines to find the length of \( v \).

To find the measure of angle \( V \) in triangle \( \triangle \mathrm{UVW} \), we use the fact that the sum of the angles in a triangle is \( 180^\circ \): \[ m \angle V = 180^\circ - m \angle W - m \angle U = 180^\circ - 79^\circ - 82^\circ = 19^\circ \]

Using the Law of Sines, we can relate the sides and angles of the triangle: \[ \frac{v}{\sin(m \angle U)} = \frac{w}{\sin(m \angle V)} \] Substituting the known values: \[ \frac{v}{\sin(82^\circ)} = \frac{690}{\sin(19^\circ)} \]

Rearranging the equation to solve for \( v \): \[ v = \frac{690 \cdot \sin(82^\circ)}{\sin(19^\circ)} \] Calculating the values gives: \[ v \approx 2098.7463 \] Rounding to the nearest tenth: \[ v \approx 2098.7 \]

Final Answer