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 \(30^\circ\), you can express the secant of that angle (\(\sec(30^\circ)\)) using the side lengths of the triangle. Here's how: 1. **Understand the Triangle Ratios:** In a \(30^\circ-60^\circ-90^\circ\) right triangle, the sides are in the ratio: \[ 1 : \sqrt{3} : 2 \] - The side opposite the \(30^\circ\) angle is the shortest and is considered to be \(1\) unit. - The side opposite the \(60^\circ\) angle is \(\sqrt{3}\) units. - The hypotenuse (opposite the \(90^\circ\) angle) is \(2\) units. 2. **Define Secant:** The secant of an angle in a right triangle is defined as the ratio of the hypotenuse to the adjacent side: \[ \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent side}} \] 3. **Apply to \(30^\circ\) Angle:** For the \(30^\circ\) angle: - **Hypotenuse:** \(2\) units - **Adjacent side:** \(\sqrt{3}\) units Therefore: \[ \sec(30^\circ) = \frac{2}{\sqrt{3}} \] 4. **Rationalize the Denominator (Optional):** To rationalize the denominator: \[ \sec(30^\circ) = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \] **Final Answer:** \[ \sec(30^\circ) = \frac{2\sqrt{3}}{3} \]