Wagner Barnett
01/04/2023 · Senior High School
Use a calculator to find a decimal approximation for the following trigonometric function. \( \sec 46^{\circ} 36^{\prime} \) \( \sec 46^{\circ} 36^{\prime} \approx \square \) (Simplify your answer. Type an integer or a decimal. Round to seven decimal places as needed.
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Step-by-step Solution
To find the decimal approximation for \( \sec 46^{\circ} 36^{\prime} \), we first need to convert the angle from degrees and minutes to decimal degrees.
1. Convert \( 36^{\prime} \) to decimal degrees:
\[
36^{\prime} = \frac{36}{60} = 0.6^{\circ}
\]
Therefore,
\[
46^{\circ} 36^{\prime} = 46^{\circ} + 0.6^{\circ} = 46.6^{\circ}
\]
2. Now, we calculate \( \sec(46.6^{\circ}) \). The secant function is the reciprocal of the cosine function:
\[
\sec(46.6^{\circ}) = \frac{1}{\cos(46.6^{\circ})}
\]
3. Using a calculator, we find \( \cos(46.6^{\circ}) \):
\[
\cos(46.6^{\circ}) \approx 0.694658
\]
4. Now, calculate \( \sec(46.6^{\circ}) \):
\[
\sec(46.6^{\circ}) \approx \frac{1}{0.694658} \approx 1.44
\]
5. Rounding to seven decimal places:
\[
\sec(46^{\circ} 36^{\prime}) \approx 1.4402026
\]
Thus, the final answer is:
\[
\sec 46^{\circ} 36^{\prime} \approx 1.4402026
\]
Quick Answer
\( \sec 46^{\circ} 36^{\prime} \approx 1.4402026 \)
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