Question
In \( \triangle K L M \), the measure of \( \angle M=90^{\circ}, M L=9, K M=40 \), and \( L K=41 \). What ratio represents the cosine of \( \angle L \) ?
Ask by Barrett Medina. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The cosine of angle \( \angle L \) is \( \frac{40}{41} \).
Solution
To find the ratio that represents the cosine of angle \( \angle L \) in triangle \( \triangle KLM \), we can use the cosine formula:
\[ \cos(\angle L) = \frac{K M}{L K} \]
Given:
- \( \angle M = 90^{\circ} \)
- \( M L = 9 \)
- \( K M = 40 \)
- \( L K = 41 \)
Substitute the given values into the cosine formula to find the ratio that represents the cosine of angle \( \angle L \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{40}{41}\)
The ratio that represents the cosine of angle \( \angle L \) in triangle \( \triangle KLM \) is \( \frac{40}{41} \) or approximately \( 0.\\dot{9}756\\dot{0} \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
In right triangle \( K L M \), to find the cosine of \( \angle L \), we use the definition of cosine, which is the ratio of the length of the adjacent side to the length of the hypotenuse. Here, the side opposite \( \angle L \) is \( M L \) (which is 9), the adjacent side is \( K M \) (which is 40), and the hypotenuse is \( L K \) (which is 41). Thus, the cosine of \( \angle L \) is calculated as follows: \[ \cos(L) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{K M}{L K} = \frac{40}{41} \] Therefore, the ratio that represents the cosine of \( \angle L \) is: \[ \cos(L) = \frac{40}{41} \]
