Q62 / 160 · TS EAMCET 2024 (10 May Shift 2) · Mathematics

If $0 \lt \theta \lt \frac{\pi}{4}$ and $8 \cos \theta+15 \sin \theta=15$, then $15 \cos \theta-8 \sin \theta=$

  • A
    15
  • B
    7
  • Correct answer
    C
    8
  • D
    23

Solution

$8 \cos \theta+15 \sin \theta=15$ $\begin{aligned} & \text { Consider }(8 \cos \theta+15 \sin \theta)^2+(15 \cos \theta-8 \sin \theta)^2 \\ & =64\left(\cos ^2 \theta+\sin ^2 \theta\right)+225\left(\cos ^2 \theta+\sin ^2 \theta\right)=289 \\ & \therefore \quad 15^2+(15 \cos \theta-8 \sin \theta)^2=289 \\ & (15 \cos \theta-8 \sin \theta)^2=64 \Rightarrow 15 \cos \theta-8 \sin \theta=8\end{aligned}$

Hence, the correct answer is option: 'C'.

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