5) In right triangle LMN with right angle at \( \mathrm{M}, \mathrm{LM}=3.28 \) and \( \mathrm{NM}=4.9 \) and \( \mathrm{LN}=5.9 \), write the ratio that represents Sin L

To find \(\sin L\) in triangle LMN, we look at the sides opposite and adjacent to angle L. In this case, the side opposite L is \(MN\), and the hypotenuse is \(LN\). Using the sides provided: \(\sin L = \frac{MN}{LN}\). First, we need to find \(MN\) using the Pythagorean theorem: \[ MN = \sqrt{LN^2 - LM^2} = \sqrt{5.9^2 - 3.28^2} \approx 4.5. \] Thus, \(\sin L = \frac{4.5}{5.9}\). So, the ratio representing \(\sin L\) is approximately \(\frac{4.5}{5.9}\). --- In the world of trigonometry, right triangles have been a cornerstone for understanding more complex mathematical concepts, such as calculus and physics. The sine function is essential in fields like engineering, architecture, and even computer graphics, where angles and distances must be calculated with precision. Just imagine how ancient builders used these principles to construct majestic structures! One common mistake when applying the sine function is confusing the opposite side with the adjacent side. It's crucial to remember that the sine of an angle in a right triangle is always the ratio of the length of the side opposite the angle to the length of the hypotenuse. To avoid errors, sketching the triangle and clearly labeling the sides can be a game-changer!