Question

1. write the fraction ratio for each of the following: (1 point)

a. \\(\tan d\\)
b. \\(\sin d\\)
c. \\(\cos d\\)

1. if \\(m\angle h = 68^{\circ}\\), what is \\(\overline{gi}\\)? round your answer to the nearest tenth. (1 point)

Explanation:

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<reasoning>

Identify sides relative to angle D in right triangle DEF

\[

$$\begin{aligned} &\text{Opposite side to } D = EF = 7\\ &\text{Adjacent side to } D = DE = 4\\ &\text{Hypotenuse} = DF = 8.1 \end{aligned}$$

\]

Calculate trigonometric ratios for angle D

\[

$$\begin{aligned} &\tan D = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{7}{4}\\ &\sin D = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{7}{8.1} = \frac{70}{81}\\ &\cos D = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{8.1} = \frac{40}{81} \end{aligned}$$

\]

Solve for GI in right triangle GHI

\[

$$\begin{aligned} &\tan H = \frac{GI}{GH} \implies \tan(68^\circ) = \frac{GI}{2}\\ &GI = 2 \cdot \tan(68^\circ) \approx 2 \cdot 2.4751 = 4.9502 \approx 5.0\text{ cm} \end{aligned}$$

\]
</reasoning>

<answer>

Question 3

a. \(\tan D = \frac{7}{4}\)

b. \(\sin D = \frac{70}{81}\)

c. \(\cos D = \frac{40}{81}\)

Question 4

\(GI = 5.0\text{ cm}\)
</answer>

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Answer:

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<reasoning>

Identify sides relative to angle D in right triangle DEF

\[

$$\begin{aligned} &\text{Opposite side to } D = EF = 7\\ &\text{Adjacent side to } D = DE = 4\\ &\text{Hypotenuse} = DF = 8.1 \end{aligned}$$

\]

Calculate trigonometric ratios for angle D

\[

$$\begin{aligned} &\tan D = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{7}{4}\\ &\sin D = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{7}{8.1} = \frac{70}{81}\\ &\cos D = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{8.1} = \frac{40}{81} \end{aligned}$$

\]

Solve for GI in right triangle GHI

\[

$$\begin{aligned} &\tan H = \frac{GI}{GH} \implies \tan(68^\circ) = \frac{GI}{2}\\ &GI = 2 \cdot \tan(68^\circ) \approx 2 \cdot 2.4751 = 4.9502 \approx 5.0\text{ cm} \end{aligned}$$

\]
</reasoning>

<answer>

Question 3

a. \(\tan D = \frac{7}{4}\)

b. \(\sin D = \frac{70}{81}\)

c. \(\cos D = \frac{40}{81}\)

Question 4

\(GI = 5.0\text{ cm}\)
</answer>

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