Law of Cosines (SSS) Score: \( 0 / 1 \) Penalty: 0.5 off Question In \( \Delta \mathrm{TUV}, \mathrm{t}=8.9 \mathrm{~cm}, u=3.2 \mathrm{~cm} \) and \( v=8.4 \mathrm{~cm} \). Find the measure of \( \angle \mathrm{U} \) to the nearest degree. Show Examples

To find the measure of \( \angle U \) in triangle \( \Delta TUV \) using the Law of Cosines, you can use the formula: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] Here, you want to find \( \angle U \), so we rearrange the formula to solve for \( \cos(U) \): \[ \cos(U) = \frac{t^2 + v^2 - u^2}{2tv} \] Plugging in the values \( t = 8.9 \, \text{cm} \), \( u = 3.2 \, \text{cm} \), and \( v = 8.4 \, \text{cm} \): \[ \cos(U) = \frac{8.9^2 + 8.4^2 - 3.2^2}{2 \cdot 8.9 \cdot 8.4} \] Calculating this gives: 1. \( 8.9^2 = 79.21 \) 2. \( 8.4^2 = 70.56 \) 3. \( 3.2^2 = 10.24 \) So, \[ \cos(U) = \frac{79.21 + 70.56 - 10.24}{2 \cdot 8.9 \cdot 8.4} \approx \frac{139.53}{149.76} \approx 0.932 \] Now, use the inverse cosine function to find \( U \): \[ U \approx \cos^{-1}(0.932) \approx 21^{\circ} \] So the measure of \( \angle U \) is approximately \( 21^{\circ} \). --- Using the Law of Cosines can be like unlocking a mystery in a triangle! This handy tool not only helps in figuring out angles when you know all three sides (SSS) but also brings clarity to a lot of real-world problems that might involve navigation, construction, or even computer graphics. It allows you to find elusive dimensions or angles, like a detective piecing together clues. When applying the Law of Cosines, one common mistake is mixing up the sides and angles in the formula. Double-check that you're inputting values correctly—using the right corresponding sides and angles helps avoid unnecessary head-scratching. Also, ensure your calculator is set to the correct mode (degrees or radians) based on your solution needs, or you might end up with some unexpected results!