Question

If $\sin\theta = \frac{3}{5}$ and $\theta<90^\circ$, then \[ \frac{\tan\theta - 2\cos\theta}{3\sin\theta + \sec\theta} = ? \]}

• A. $-\frac{17}{61}$
• B. $61$
• C. $\frac{61}{17}$
• D. $71$

Hint:

For $\sin\theta = \frac{3}{5}$, always recall the $3$-$4$-$5$ triangle.

Answer 1

The Correct Option is A

Concept: Use Pythagoras and trigonometric identities.
Step 1: Find $\cos\theta$.
\[ \sin\theta = \frac{3}{5} \Rightarrow \cos\theta = \frac{4}{5} \]
Step 2: Find $\tan\theta$ and $\sec\theta$.
\[ \tan\theta = \frac{3}{4},\quad \sec\theta = \frac{5}{4} \]
Step 3: Substitute.
\[ \frac{\frac{3}{4} - 2\cdot \frac{4}{5}}{3\cdot \frac{3}{5} + \frac{5}{4}} \] \[ = \frac{\frac{3}{4} - \frac{8}{5}}{\frac{9}{5} + \frac{5}{4}} \]
Step 4: Simplify numerator.
\[ \frac{15 - 32}{20} = \frac{-17}{20} \]
Step 5: Simplify denominator.
\[ \frac{36 + 25}{20} = \frac{61}{20} \]
Step 6: Final value.
\[ \frac{-17}{20} \div \frac{61}{20} = -\frac{17}{61} \]
Hence, the answer is $-\frac{17{61}$.