The angle \( \theta \) is in Quadrant I with \( \sec (\theta)=\frac{5}{3} \). Which equation could be used to find \( \tan (\theta) \) ? \( \begin{array}{l}\frac{5}{3}-\tan (\theta)=1 \\ 1-\left(\frac{5}{3}\right)^{2}+\tan ^{2}(\theta)=1 \\ 1+\tan ^{2}(\theta) \\ 1+\tan ^{2}(\theta)=\left(\frac{5}{3}\right)^{2}\end{array} \)

To find \( \tan(\theta) \) given \( \sec(\theta) = \frac{5}{3} \), you can use the identity that relates secant and tangent: Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we have \( \cos(\theta) = \frac{3}{5} \). Using the Pythagorean identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \), we can square \( \sec(\theta) \): \[ 1 + \tan^2(\theta) = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \] This leads us directly to the equation: \[ 1 + \tan^2(\theta) = \frac{25}{9} \] Thus, the correct equation that could be used to find \( \tan(\theta) \) is: \( 1 + \tan^2(\theta) = \left( \frac{5}{3} \right)^2 \). Now, tackling the historical aspect of tangent, it dates back to ancient civilizations. The term "tangent" was first introduced by the mathematician "Bartholomaeus Pitiscus" in the early 17th century when he published a book on trigonometry. He coined it from the Latin word "tangens," meaning "touching," as it refers to the line that touches the circle at one point. If you're interested in applying what you've learned in the real world, think of how the tangent function is crucial in fields such as engineering and physics. It's used to calculate angles of elevation and depression, especially in scenarios like determining the height of a building or troubleshooting optics. It's amazing how these concepts extend beyond mere formulas!