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:

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<reasoning>

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 \(JL = 13\).

\[
\cos(L) = \frac{5}{13}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{5}{13}\)</mcq-correct>
<mcq-option>(B) \(\frac{5}{12}\)</mcq-option>
<mcq-option>(C) \(\frac{12}{5}\)</mcq-option>
<mcq-option>(D) \(\frac{12}{13}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

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 \(JL = 13\).

\[
\cos(L) = \frac{5}{13}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{5}{13}\)</mcq-correct>
<mcq-option>(B) \(\frac{5}{12}\)</mcq-option>
<mcq-option>(C) \(\frac{12}{5}\)</mcq-option>
<mcq-option>(D) \(\frac{12}{13}\)</mcq-option>
</answer>

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