Question

directions: find the indicated trigonometric ratio as a fraction in simplest form.

1. \\( \sin l = \\)
2. \\( \tan n = \\)
3. \\( \cos l = \\)
4. \\( \sin n = \\)

Explanation:

Step1: Find the length of side \(LM\) using the Pythagorean theorem

In right - triangle \(LMN\) with \(LN = 10\) (hypotenuse) and \(MN=6\) (one leg). Let \(LM=x\). By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = LN\), \(a = LM\), \(b = MN\). So \(x^{2}+6^{2}=10^{2}\), \(x^{2}=100 - 36=64\), \(x = 8\).

Step2: Calculate \(\sin L\)

The sine of an angle in a right - triangle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle L\), the opposite side is \(MN = 6\) and the hypotenuse is \(LN = 10\). So \(\sin L=\frac{MN}{LN}=\frac{6}{10}=\frac{3}{5}\).

Step3: Calculate \(\tan N\)

The tangent of an angle in a right - triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\angle N\), the opposite side is \(LM = 8\) and the adjacent side is \(MN = 6\). So \(\tan N=\frac{LM}{MN}=\frac{8}{6}=\frac{4}{3}\).

Step4: Calculate \(\cos L\)

The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\angle L\), the adjacent side is \(LM = 8\) and the hypotenuse is \(LN = 10\). So \(\cos L=\frac{LM}{LN}=\frac{8}{10}=\frac{4}{5}\).

Step5: Calculate \(\sin N\)

The sine of an angle in a right - triangle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle N\), the opposite side is \(LM = 8\) and the hypotenuse is \(LN = 10\). So \(\sin N=\frac{LM}{LN}=\frac{8}{10}=\frac{4}{5}\).

Answer:

1. \(\frac{3}{5}\)
2. \(\frac{4}{3}\)
3. \(\frac{4}{5}\)
4. \(\frac{4}{5}\)