If \(\tan A=\dfrac{3}{4}\), prove that \(\sin A\cos A=\dfrac{12}{25}\).
\(\displaystyle \sin A\cos A=\dfrac{12}{25}\).
Step 1: Recall the meaning of \(\tan A\). It is defined as:
\(\tan A = \dfrac{\text{opposite side}}{\text{adjacent side}}\).
Step 2: Here, \(\tan A = \dfrac{3}{4}\). This means the length of the opposite side = 3 units, and the length of the adjacent side = 4 units.
Step 3: To find the hypotenuse of the right triangle, use the Pythagoras theorem:
\(\text{Hypotenuse} = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2}\).
\(= \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
Step 4: Now we can find sine and cosine:
Step 5: Multiply sine and cosine:
\(\sin A \cdot \cos A = \dfrac{3}{5} \times \dfrac{4}{5} = \dfrac{12}{25}\).
Final Answer: \(\sin A \cos A = \dfrac{12}{25}\).