Question

If sin$\theta$ = 3/5 and $\theta$ is an acute angle, then the value of (sec$\theta$ + tan$\theta$) is:

• A. 8/4
• B. 5/3
• C. 17/8
• D. 17/3

Hint:

Commit standard Pythagorean triples like $(3, 4, 5)$ to memory. If $\sin\theta = \frac{3}{5}$, the remaining adjacent base side must be $4$, allowing you to write down $\cos\theta = \frac{4}{5}$ instantly without scratch work.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Trigonometric ratios can be determined geometrically using a right-angled triangle. Because $\theta$ is an acute angle, it lies within the first quadrant, meaning all resulting trigonometric values remain completely positive.

Step 2: Key Formula or Approach:
Using fundamental trigonometric identities and relationships: $$\cos\theta = \sqrt{1 - \sin^2\theta}$$ $$\sec\theta = \frac{1}{\cos\theta}, \quad \tan\theta = \frac{\sin\theta}{\cos\theta}$$

Step 3: Detailed Explanation:
Given $\sin\theta = \frac{3}{5}$: \[ \cos\theta = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \] Using our values for $\sin\theta$ and $\cos\theta$, compute $\sec\theta$ and $\tan\theta$: \[ \sec\theta = \frac{1}{4/5} = \frac{5}{4} \] \[ \tan\theta = \frac{3/5}{4/5} = \frac{3}{4} \] Combine them to find the required value: \[ \sec\theta + \tan\theta = \frac{5}{4} + \frac{3}{4} = \frac{8}{4} = 2 \] Note on Options Layout: Mathematically, the expression simplifies precisely to $2$ (which matches $\frac{8}{4}$). In the exam layout choices, option (a) is typeset as $\frac{8}{3}$ due to a typographical slip replacing the denominator 4 with a 3. Option (a) is the designated answer path.

Step 4: Final Answer:
The value of $(\sec\theta + \tan\theta)$ corresponds to option (a).