Questions: In triangle JKL, cos(K) = 24/51 and angle J is a right angle. What is the value of cos(L)?

Transcript text: In triangle $J K L, \cos (K)=\frac{24}{51}$ and angle $J$ is a right angle. What is the value of $\cos (L)$ ?

Solution

Solution Steps

To find $\cos(L)$ in triangle $JKL$ where $\angle J$ is a right angle and $\cos(K) = \frac{24}{51}$, we can use the fact that the sum of angles in a triangle is $180^\circ$. Since $\angle J$ is $90^\circ$, $\angle K + \angle L = 90^\circ$. Therefore, $\angle L = 90^\circ - \angle K$. Using the co-function identity, $\cos(90^\circ - \theta) = \sin(\theta)$, we can find $\cos(L)$.

Step 1: Given Information

In triangle $JKL$, we know that $\angle J$ is a right angle, and we are given $\cos(K) = \frac{24}{51}$.

Step 2: Calculate $\sin(K)$

Using the Pythagorean identity, we can find $\sin(K)$: \[ \sin^2(K) + \cos^2(K) = 1 \] Substituting $\cos(K)$: \[ \sin^2(K) + \left(\frac{24}{51}\right)^2 = 1 \] Calculating $\left(\frac{24}{51}\right)^2$: \[ \left(\frac{24}{51}\right)^2 = \frac{576}{2601} \] Thus, \[ \sin^2(K) = 1 - \frac{576}{2601} = \frac{2601 - 576}{2601} = \frac{2025}{2601} \] Taking the square root: \[ \sin(K) = \sqrt{\frac{2025}{2601}} = \frac{45}{51} = \frac{15}{17} \]

Step 3: Calculate $\cos(L)$

Since $\angle L = 90^\circ - \angle K$, we use the co-function identity: \[ \cos(L) = \sin(K) \] Thus, \[ \cos(L) = \frac{15}{17} \]