Question

find the exact value of sec 30°. sec 30° = \\(\square\\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the definition of secant

The secant of an angle in a right triangle is defined as the reciprocal of the cosine of that angle, so \(\sec\theta=\frac{1}{\cos\theta}\).

Step2: Find the value of \(\cos30^{\circ}\)

We know from the special right - triangle (30 - 60 - 90 triangle) or the unit - circle definition that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\).

Step3: Calculate \(\sec30^{\circ}\)

Since \(\sec30^{\circ}=\frac{1}{\cos30^{\circ}}\), substitute \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\) into the formula. We get \(\sec30^{\circ}=\frac{1}{\frac{\sqrt{3}}{2}}\). When dividing by a fraction, we multiply by its reciprocal, so \(\frac{1}{\frac{\sqrt{3}}{2}} = 1\times\frac{2}{\sqrt{3}}=\frac{2}{\sqrt{3}}\). To rationalize the denominator, we multiply the numerator and denominator by \(\sqrt{3}\): \(\frac{2\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{2\sqrt{3}}{3}\).

Answer:

\(\frac{2\sqrt{3}}{3}\)