Question

use sine, cosine, or tangent to solve each right triangle. round all angle measures to the nearest tenth and all side measures to the nearest tenth.

\\(m\angle t = \\)
\\(rt = \\)
\\(rs = \\)

Explanation:

<pre_analysis>
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</pre_analysis>

<reasoning>

Find the measure of angle T

\[

$$\begin{aligned} m\angle T &= 90^\circ - 57^\circ \\ m\angle T &= 33^\circ \end{aligned}$$

\]

Calculate the length of RT

\[

$$\begin{aligned} \sin(57^\circ) &= \frac{15}{RT} \\ RT &= \frac{15}{\sin(57^\circ)} \\ RT &\approx \frac{15}{0.83867} \approx 17.885 \\ RT &\approx 17.9 \end{aligned}$$

\]

Calculate the length of RS

\[

$$\begin{aligned} \tan(57^\circ) &= \frac{15}{RS} \\ RS &= \frac{15}{\tan(57^\circ)} \\ RS &\approx \frac{15}{1.53986} \approx 9.741 \\ RS &\approx 9.7 \end{aligned}$$

\]
</reasoning>

<answer>
Use sine, cosine, or tangent to solve each right triangle. Round all angle measures to the nearest tenth and all side measures to the nearest tenth.

\(m\angle T =\) <blank>33</blank>\(^\circ\)

\(RT =\) <blank>17.9</blank>

\(RS =\) <blank>9.7</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Mathematics",
"Geometry",
"Trigonometric Ratios"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Find the measure of angle T

\[

$$\begin{aligned} m\angle T &= 90^\circ - 57^\circ \\ m\angle T &= 33^\circ \end{aligned}$$

\]

Calculate the length of RT

\[

$$\begin{aligned} \sin(57^\circ) &= \frac{15}{RT} \\ RT &= \frac{15}{\sin(57^\circ)} \\ RT &\approx \frac{15}{0.83867} \approx 17.885 \\ RT &\approx 17.9 \end{aligned}$$

\]

Calculate the length of RS

\[

$$\begin{aligned} \tan(57^\circ) &= \frac{15}{RS} \\ RS &= \frac{15}{\tan(57^\circ)} \\ RS &\approx \frac{15}{1.53986} \approx 9.741 \\ RS &\approx 9.7 \end{aligned}$$

\]
</reasoning>

<answer>
Use sine, cosine, or tangent to solve each right triangle. Round all angle measures to the nearest tenth and all side measures to the nearest tenth.

\(m\angle T =\) <blank>33</blank>\(^\circ\)

\(RT =\) <blank>17.9</blank>

\(RS =\) <blank>9.7</blank>
</answer>

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"Mathematics",
"Geometry",
"Trigonometric Ratios"
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