Question

given \\(\delta def\\), which is not equal to \\(\cos(f)\\)?

\\(\sin(f)\\)

\\(\sin(d)\\)

\\(\cos(d)\\)

\\(\tan(f)\\)

Explanation:

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<reasoning>

Identify the given triangle properties

We are given a right isosceles triangle \(\triangle DEF\) with:

• Right angle at vertex \(E\) (\(\angle E = 90^\circ\))
• Side lengths \(DE = 5\) and \(EF = 5\)
• Hypotenuse \(DF = 5\sqrt{2}\)

Calculate the target trigonometric ratio

Using the Cosine Ratio knowledge point
\[
\cos(F) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{EF}{DF} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]

Evaluate each option ratio

Using the Trigonometric Ratios knowledge point
\[

$$\begin{aligned} &\sin(F) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{DE}{DF} = \frac{5}{5\sqrt{2}} = \frac{\sqrt{2}}{2}\\ &\sin(D) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{EF}{DF} = \frac{5}{5\sqrt{2}} = \frac{\sqrt{2}}{2}\\ &\cos(D) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{DE}{DF} = \frac{5}{5\sqrt{2}} = \frac{\sqrt{2}}{2}\\ &\tan(F) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{DE}{EF} = \frac{5}{5} = 1 \end{aligned}$$

\]

Determine the non-equal ratio

Comparing the calculated values:

• \(\cos(F) = \frac{\sqrt{2}}{2}\)
• \(\tan(F) = 1

eq \frac{\sqrt{2}}{2}\)
</reasoning>

<answer>
<mcq-option>(A) \(\sin (F)\)</mcq-option>
<mcq-option>(B) \(\sin (D)\)</mcq-option>
<mcq-option>(C) \(\cos (D)\)</mcq-option>
<mcq-correct>(D) \(\tan (F)\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the given triangle properties

We are given a right isosceles triangle \(\triangle DEF\) with:

• Right angle at vertex \(E\) (\(\angle E = 90^\circ\))
• Side lengths \(DE = 5\) and \(EF = 5\)
• Hypotenuse \(DF = 5\sqrt{2}\)

Calculate the target trigonometric ratio

Using the Cosine Ratio knowledge point
\[
\cos(F) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{EF}{DF} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]

Evaluate each option ratio

Using the Trigonometric Ratios knowledge point
\[

$$\begin{aligned} &\sin(F) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{DE}{DF} = \frac{5}{5\sqrt{2}} = \frac{\sqrt{2}}{2}\\ &\sin(D) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{EF}{DF} = \frac{5}{5\sqrt{2}} = \frac{\sqrt{2}}{2}\\ &\cos(D) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{DE}{DF} = \frac{5}{5\sqrt{2}} = \frac{\sqrt{2}}{2}\\ &\tan(F) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{DE}{EF} = \frac{5}{5} = 1 \end{aligned}$$

\]

Determine the non-equal ratio

Comparing the calculated values:

• \(\cos(F) = \frac{\sqrt{2}}{2}\)
• \(\tan(F) = 1

eq \frac{\sqrt{2}}{2}\)
</reasoning>

<answer>
<mcq-option>(A) \(\sin (F)\)</mcq-option>
<mcq-option>(B) \(\sin (D)\)</mcq-option>
<mcq-option>(C) \(\cos (D)\)</mcq-option>
<mcq-correct>(D) \(\tan (F)\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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"Trigonometric Ratios"
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