Question

1. express tan l as a fraction in simplest terms.

express tan o as a fraction in simplest terms.

1. find the value of sin n rounded to the nearest hundredth, if necessary.

1. express sin m as a fraction in simplest terms.

Explanation:

Step1: Find side JK for tan L

First, calculate side $JK$ using Pythagoras:
$$JK = \sqrt{(\sqrt{73})^2 - 6^2} = \sqrt{73 - 36} = \sqrt{37}$$
$\tan L = \frac{\text{opposite}}{\text{adjacent}} = \frac{JK}{LK} = \frac{\sqrt{37}}{6}$

Step2: Simplify tan O

$\tan O = \frac{\text{opposite}}{\text{adjacent}} = \frac{MN}{ON} = \frac{25}{10} = \frac{5}{2}$

Step3: Calculate sin N

$\sin N = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{\sqrt{85}} \approx 0.76$

Step4: Simplify sin M

$\sin M = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{87}}{13}$

Answer:

1. $\frac{\sqrt{37}}{6}$
2. $\frac{5}{2}$
3. $0.76$
4. $\frac{\sqrt{87}}{13}$