In a right triangle where one angle measures 30 degrees, how do you express the secant of that angle?

In a right triangle with one angle measuring 30 degrees, you can express the **secant** of that angle by following these steps: 1. **Understand Secant:** The secant of an angle in a right triangle is the reciprocal of the cosine of that angle. \[ \sec(\theta) = \frac{1}{\cos(\theta)} \] 2. **Identify the Triangle Ratios:** A 30-60-90 right triangle has sides in the ratio of \(1 : \sqrt{3} : 2\), where: - The side opposite the 30° angle is \(1\) (shortest side). - The side opposite the 60° angle is \(\sqrt{3}\). - The hypotenuse opposite the 90° angle is \(2\). 3. **Calculate Cosine of 30 Degrees:** Cosine is the ratio of the adjacent side to the hypotenuse. For the 30° angle: \[ \cos(30°) = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{\sqrt{3}}{2} \] 4. **Find Secant of 30 Degrees:** Using the reciprocal relationship: \[ \sec(30°) = \frac{1}{\cos(30°)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \] To rationalize the denominator: \[ \sec(30°) = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \] **Final Expression:** \[ \sec(30°) = \frac{2\sqrt{3}}{3} \] This means that in a 30-60-90 right triangle, the secant of the 30-degree angle is \(\frac{2\sqrt{3}}{3}\).