Pregunta
Find the exact value of \( \sec 30^{\circ} \) sec \( 30^{\circ}=\square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.
Ask by Barker Henry. in the United States
Feb 03,2025
Solución de tutoría real
Respuesta verificada por el tutor
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\[
\sec 30^{\circ} = \frac{2\,\sqrt{3}}{3}
\]
Solución
To find the exact value of \( \sec 30^{\circ} \), we can use the relationship between secant and cosine:
\[
\sec \theta = \frac{1}{\cos \theta}
\]
First, find \( \cos 30^{\circ} \):
\[
\cos 30^{\circ} = \frac{\sqrt{3}}{2}
\]
Now, take the reciprocal to find \( \sec 30^{\circ} \):
\[
\sec 30^{\circ} = \frac{1}{\cos 30^{\circ}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}
\]
To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{3} \):
\[
\sec 30^{\circ} = \frac{2 \sqrt{3}}{3}
\]
**Answer:**
\[
\sec 30^{\circ} = \frac{2\,\sqrt{3}}{3}
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Mind Expander
To find the exact value of \( \sec 30^{\circ} \), we start by recalling that the secant function is the reciprocal of the cosine function. Thus, \[ \sec 30^{\circ} = \frac{1}{\cos 30^{\circ}}. \] Next, we know from trigonometric ratios that \[ \cos 30^{\circ} = \frac{\sqrt{3}}{2}. \] Now, substituting this value back into our equation gives us \[ \sec 30^{\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}. \] To simplify, we rationalize the denominator: \[ \sec 30^{\circ} = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}. \] Thus, the exact value is \[ \sec 30^{\circ} = \frac{2\sqrt{3}}{3}. \]
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