Question

find the exact value of sec 30°

sec 30°=
(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, \(\sec\theta=\frac{1}{\cos\theta}\). For \(\theta = 30^{\circ}\), we have \(\sec30^{\circ}=\frac{1}{\cos30^{\circ}}\).

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

We know from the special right triangles (or the unit circle) that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\).

Step3: Substitute the value of \(\cos30^{\circ}\) into the secant formula

Substituting \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\) into \(\sec30^{\circ}=\frac{1}{\cos30^{\circ}}\), we get \(\sec30^{\circ}=\frac{1}{\frac{\sqrt{3}}{2}}\).
To divide by a fraction, we multiply by its reciprocal, so \(\frac{1}{\frac{\sqrt{3}}{2}} = 1\times\frac{2}{\sqrt{3}}=\frac{2}{\sqrt{3}}\).

Step4: Rationalize the denominator

To rationalize the denominator of \(\frac{2}{\sqrt{3}}\), we multiply the numerator and the denominator by \(\sqrt{3}\): \(\frac{2\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{2\sqrt{3}}{3}\).

Answer:

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