Question

Find sinL, cosL, tanL, sinM, cosM, and tanM when ℓ=15, m=36, and n=39. Match each ratio to the corresponding trigonometric expression.

sinL=
cosL=
tanL=
sinM=
cosM=
tanM=

Answer 1

To find the trigonometric ratios for angles \( L \) and \( M \) given \( \ell = 15 \), \( m = 36 \), and \( n = 39 \), we first need to determine the angles \( L \) and \( M \) for triangles formed with these sides.

Given the lengths of the sides, we can consider that:
1. \( \ell = 15 \) could be the opposite side to angle \( L \),
2. \( m = 36 \) could be the adjacent side to angle \( L \),
3. \( n = 39 \) is the hypotenuse.

We can start with calculating the ratios for angle \( L \):

### Calculating for Angle \( L \):
1. **Sine**:
\[
\sin L = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\ell}{n} = \frac{15}{39}
\]
Simplifying:
\[
\sin L = \frac{5}{13}
\]

2. **Cosine**:
\[
\cos L = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{m}{n} = \frac{36}{39}
\]
Simplifying:
\[
\cos L = \frac{12}{13}
\]

3. **Tangent**:
\[
\tan L = \frac{\text{opposite}}{\text{adjacent}} = \frac{\ell}{m} = \frac{15}{36}
\]
Simplifying:
\[
\tan L = \frac{5}{12}
\]

### Calculating for Angle \( M \):
Now we can calculate for angle \( M \). Angle \( M \) is the angle opposite \( m \):

1. **Sine**:
\[
\sin M = \frac{m}{n} = \frac{36}{39}
\]
Simplifying:
\[
\sin M = \frac{12}{13}
\]

2. **Cosine**:
\[
\cos M = \frac{\ell}{n} = \frac{15}{39}
\]
Simplifying:
\[
\cos M = \frac{5}{13}
\]

3. **Tangent**:
\[
\tan M = \frac{m}{\ell} = \frac{36}{15}
\]
Simplifying:
\[
\tan M = \frac{12}{5}
\]

### Summary:
Now, we can summarize the results:

- \(\sin L = \frac{5}{13}\)
- \(\cos L = \frac{12}{13}\)
- \(\tan L = \frac{5}{12}\)
- \(\sin M = \frac{12}{13}\)
- \(\cos M = \frac{5}{13}\)
- \(\tan M = \frac{12}{5}\)

Thus, the final results are:
- \( \sin L = \frac{5}{13} \)
- \( \cos L = \frac{12}{13} \)
- \( \tan L = \frac{5}{12} \)
- \( \sin M = \frac{12}{13} \)
- \( \cos M = \frac{5}{13} \)
- \( \tan M = \frac{12}{5} \)