An angle \(\theta\) obeys the relation \(8\sin\theta+15\cos\theta=17\). Which value does \(\tan\theta\) take?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{8}{15}\)
Pair the given expression with \(15\sin\theta-8\cos\theta\). When both are squared and added, the cross terms cancel and \(\sin^2\theta+\cos^2\theta=1\) leaves \(8^2+15^2=289=17^2\).
\((8\sin\theta+15\cos\theta)^2+(15\sin\theta-8\cos\theta)^2=(64+225)(\sin^2\theta+\cos^2\theta)=289\)
\(17^2+(15\sin\theta-8\cos\theta)^2=289\), so \(15\sin\theta-8\cos\theta=0\)
\(15\sin\theta=8\cos\theta\), so \(\tan\theta=\displaystyle\frac{8}{15}\)
Check: the angle with \(\sin\theta=\frac{8}{17}\) and \(\cos\theta=\frac{15}{17}\) has \(\tan\theta=\frac{8}{15}\), and \(8\times\frac{8}{17}+15\times\frac{15}{17}=\frac{64+225}{17}=17\).