Question

given right triangle jkl, what is the value of \\(\cos(l)\\)?

\\(\frac{5}{13}\\)

\\(\frac{5}{12}\\)

\\(\frac{12}{5}\\)

\\(\frac{12}{13}\\)

Explanation:

Identify given values in the right triangle

We are given right triangle \(JKL\) with:

• Right angle at vertex \(K\).
• Side \(JK = 12\) (opposite to angle \(L\)).
• Side \(KL = 5\) (adjacent to angle \(L\)).

Calculate the hypotenuse using the Pythagorean theorem

To find the hypotenuse \(JL\), we apply:

$$ JL^2 = JK^2 + KL^2 $$

$$ JL^2 = 12^2 + 5^2 = 144 + 25 = 169 $$

$$ JL = \sqrt{169} = 13 $$

Apply the definition of the cosine ratio

The cosine of angle \(L\) is defined as:

$$ \cos(L) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$

Using our values:

• Adjacent side to \(L\) is \(KL = 5\).
• Hypotenuse is \(JL = 13\).

$$ \cos(L) = \frac{5}{13} $$

Answer:

• (A) \(\frac{5}{13}\) (Correct answer)
• (B) \(\frac{5}{12}\)
• (C) \(\frac{12}{5}\)
• (D) \(\frac{12}{13}\)