An angle \(\theta\) obeys the relation \(7\sin\theta+24\cos\theta=25\). Which value does \(\tan\theta\) take?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{7}{24}\)
Pair the given expression with \(24\sin\theta-7\cos\theta\). When both are squared and added, the cross terms cancel and \(\sin^2\theta+\cos^2\theta=1\) leaves \(7^2+24^2=625=25^2\).
\((7\sin\theta+24\cos\theta)^2+(24\sin\theta-7\cos\theta)^2=(49+576)(\sin^2\theta+\cos^2\theta)=625\)
\(25^2+(24\sin\theta-7\cos\theta)^2=625\), so \(24\sin\theta-7\cos\theta=0\)
\(24\sin\theta=7\cos\theta\), so \(\tan\theta=\displaystyle\frac{7}{24}\)
Check: the angle with \(\sin\theta=\frac{7}{25}\) and \(\cos\theta=\frac{24}{25}\) has \(\tan\theta=\frac{7}{24}\), and \(7\times\frac{7}{25}+24\times\frac{24}{25}=\frac{49+576}{25}=25\).