Question

If \(3\tan A = 4 \quad (0^\circ < A < 90^\circ)\), then the value of \(\sin A\) is \dots

• A. \(\frac{3}{5}\)
• B. \(\frac{4}{5}\)
• C. \(\frac{3}{4}\)
• D. \(\frac{5}{4}\)

Hint:

Whenever: \[ \tan A = \frac{a}{b} \] take: \[ \text{Perpendicular} = a,\quad \text{Base} = b \] then use Pythagoras theorem to find the hypotenuse quickly.

Answer 1

The Correct Option is B

Concept: The trigonometric ratio: \[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} \] Using this ratio, we can construct a right triangle and then use the Pythagoras Theorem to find the hypotenuse. Finally: \[ \sin A = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]

Step 1: Simplify the given equation. Given: \[ 3\tan A = 4 \] Divide both sides by 3: \[ \tan A = \frac{4}{3} \]

Step 2: Interpret the tangent ratio. Since: \[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} \] we assume: \[ \text{Perpendicular} = 4 \] and: \[ \text{Base} = 3 \]

Step 3: Draw and analyze the right triangle. The triangle now has:
• perpendicular side \(= 4\)
• base side \(= 3\)
• hypotenuse \(= h\) Using Pythagoras Theorem: \[ h^2 = 4^2 + 3^2 \] \[ h^2 = 16 + 9 \] \[ h^2 = 25 \]

Step 4: Find the hypotenuse. \[ h = \sqrt{25} \] \[ h = 5 \]

Step 5: Calculate \(\sin A\). By definition: \[ \sin A = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \] Substitute values: \[ \sin A = \frac{4}{5} \]

Step 6: Verify the answer. Since: \[ 0^\circ < A < 90^\circ \] all trigonometric ratios are positive. Therefore: \[ \frac{4}{5} \] is valid. Final Answer: \[ \boxed{\frac{4}{5}} \]