Question

question 1

find the length of the adjacent side of a right-angled triangle if the opposite side is \\(5\text{ cm}\\) and the angle is \\(60^\circ\\), using the cosine function and exact values.

• i dont know
• \\(4.5\text{ cm}\\)
• \\(3.5\text{ cm}\\)
• \\(2.5\text{ cm}\\)
• \\(5.5\text{ cm}\\)
• \\(1.5\text{ cm}\\)

Explanation:

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<reasoning>

Find the hypotenuse using the sine function

\[

$$\begin{aligned} &\sin(60^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} \\ &\frac{\sqrt{3}}{2} = \frac{5}{H} \\ &H = \frac{10}{\sqrt{3}} \approx 5.7735\text{ cm} \end{aligned}$$

\]

Find the adjacent side using the cosine function

\[

$$\begin{aligned} &\cos(60^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \\ &\frac{1}{2} = \frac{A}{\frac{10}{\sqrt{3}}} \\ &A = \frac{5}{\sqrt{3}} \approx 2.8867\text{ cm} \end{aligned}$$

\]

Identify the closest option or direct tangent relationship

\[

$$\begin{aligned} &\tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} \\ &\sqrt{3} = \frac{5}{A} \\ &A = \frac{5}{\sqrt{3}} \approx 2.8867\text{ cm} \approx 2.9\text{ cm} \end{aligned}$$

\]
Note: The question specifically requests "using the cosine function and exact values" to find the adjacent side, which leads to \(2.9\text{ cm}\). Looking at the options, \(2.5\text{ cm}\) is the closest provided numerical option.
</reasoning>

<answer>
<mcq-option>I don't know</mcq-option>
<mcq-option>4.5 cm</mcq-option>
<mcq-option>3.5 cm</mcq-option>
<mcq-correct>2.5 cm</mcq-correct>
<mcq-option>5.5 cm</mcq-option>
<mcq-option>1.5 cm</mcq-option>
</answer>

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Answer:

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<reasoning>

Find the hypotenuse using the sine function

\[

$$\begin{aligned} &\sin(60^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} \\ &\frac{\sqrt{3}}{2} = \frac{5}{H} \\ &H = \frac{10}{\sqrt{3}} \approx 5.7735\text{ cm} \end{aligned}$$

\]

Find the adjacent side using the cosine function

\[

$$\begin{aligned} &\cos(60^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \\ &\frac{1}{2} = \frac{A}{\frac{10}{\sqrt{3}}} \\ &A = \frac{5}{\sqrt{3}} \approx 2.8867\text{ cm} \end{aligned}$$

\]

Identify the closest option or direct tangent relationship

\[

$$\begin{aligned} &\tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} \\ &\sqrt{3} = \frac{5}{A} \\ &A = \frac{5}{\sqrt{3}} \approx 2.8867\text{ cm} \approx 2.9\text{ cm} \end{aligned}$$

\]
Note: The question specifically requests "using the cosine function and exact values" to find the adjacent side, which leads to \(2.9\text{ cm}\). Looking at the options, \(2.5\text{ cm}\) is the closest provided numerical option.
</reasoning>

<answer>
<mcq-option>I don't know</mcq-option>
<mcq-option>4.5 cm</mcq-option>
<mcq-option>3.5 cm</mcq-option>
<mcq-correct>2.5 cm</mcq-correct>
<mcq-option>5.5 cm</mcq-option>
<mcq-option>1.5 cm</mcq-option>
</answer>

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