An angle \(\theta\) lying strictly between \(0^\circ\) and \(90^\circ\) is chosen so that \(\sin\theta+\cos\theta=\displaystyle\frac{7}{5}\). Work out the product \(\sin\theta\cos\theta\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{12}{25}\)
Squaring \(\sin\theta+\cos\theta\) gives \(\sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta\), and the first two terms add up to 1. The product is then found directly.
\((\sin\theta+\cos\theta)^2=\displaystyle\frac{49}{25}\)
\(1+2\sin\theta\cos\theta=\displaystyle\frac{49}{25}\), so \(2\sin\theta\cos\theta=\displaystyle\frac{24}{25}\)
\(\sin\theta\cos\theta=\displaystyle\frac{12}{25}\)
Such an angle exists: \(\sin\theta=\frac{3}{5}\), \(\cos\theta=\frac{4}{5}\) (or the other way round) fit the condition.