Step 1 : Understanding the Question:
This question is rooted in the basics of trigonometry and right-angled triangle properties. We are given the sine ratio of an acute angle $\theta$. Since the angle is acute ($0^\circ<\theta<90^\circ$), it lies in the first quadrant where all trigonometric ratios (sine, cosine, tangent, etc.) are positive. Our task is to determine the sum of the secant and tangent of this same angle. This requires us to first find the remaining sides of the triangle or use trigonometric identities to derive the cosine value, which is the foundation for both secant and tangent.
Step 2 : Key Formulas and approach:
We can use the Pythagorean identity or the side-ratio definition of a right triangle:
$\sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$.
Pythagorean Theorem: $Base^2 + Perpendicular^2 = Hypotenuse^2$.
$\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}$.
$\sec \theta = \frac{1}{\cos \theta}$ and $\tan \theta = \frac{\sin \theta}{\cos \theta}$.
The approach is to identify the sides of the triangle, find the missing side, and then calculate the required expression.
Step 3 : Detailed Explanation:
Given $\sin \theta = \frac{3}{5}$. This implies Perpendicular ($P$) = 3 and Hypotenuse ($H$) = 5.
Using the Pythagorean triple (3, 4, 5) or calculating via the theorem: $B = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$.
Now, we can find $\cos \theta = \frac{Base}{Hypotenuse} = \frac{4}{5}$.
Calculate $\sec \theta$: Since $\sec \theta$ is the reciprocal of $\cos \theta$, $\sec \theta = \frac{5}{4}$.
Calculate $\tan \theta$: Since $\tan \theta = \frac{Perpendicular}{Base}$, $\tan \theta = \frac{3}{4}$.
Now, find the sum: $(\sec \theta + \tan \theta) = \frac{5}{4} + \frac{3}{4} = \frac{8}{4} = 2$.
Step 4 : Final Answer:
The value of $(\sec \theta + \tan \theta)$ is 2 or 8/4 (Option A).