Question

find uv.
write your answer as an integer or as a decimal rounded to the nearest tenth.
uv =

Explanation:

Step1: Identify the triangle type and trigonometric ratio

We have a right - triangle \(UVT\) with \(\angle V = 90^{\circ}\), \(\angle T=49^{\circ}\) and hypotenuse \(UT = 8\). We want to find the length of \(UV\). In a right - triangle, the cosine of an angle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\angle T = 49^{\circ}\), the adjacent side to \(\angle T\) is \(TV\) and the opposite side is \(UV\)? Wait, no. Wait, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle T\), the side opposite to \(\angle T\) is \(UV\) and the hypotenuse is \(UT = 8\). So \(\sin(49^{\circ})=\frac{UV}{UT}\)

Step2: Solve for \(UV\)

Since \(UT = 8\) and \(\sin(49^{\circ})=\frac{UV}{8}\), we can re - arrange the formula to get \(UV=8\times\sin(49^{\circ})\)

We know that \(\sin(49^{\circ})\approx0.7547\) (using a calculator). Then \(UV = 8\times0.7547=6.0376\approx6.0\) (rounded to the nearest tenth)

Answer:

\(6.0\)