If \(\sin A=\frac{4}{5}\), then \((3-\tan A)(2+\cos A)=\)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{13}{3}\)
Take A as an angle of a right triangle (an acute angle). \(\sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{4}{5}\), so the third side comes from Pythagoras’ theorem, and \(\cos A\) and \(\tan A\) follow.
Opposite side 4 and hypotenuse 5, so the adjacent side \(=\sqrt{5^{2}-4^{2}}=3\)
Adjacent over hypotenuse and opposite over adjacent: \(\displaystyle \cos A=\frac{3}{5}\), \(\displaystyle \tan A=\frac{4}{3}\)
The first bracket: \(\displaystyle 3-\tan A=3-\frac{4}{3}=\frac{5}{3}\)
The second bracket: \(\displaystyle 2+\cos A=2+\frac{3}{5}=\frac{13}{5}\)
\(\displaystyle (3-\tan A)(2+\cos A)=\frac{5}{3}\times\frac{13}{5}=\frac{13}{3}\)