Verified by Ajay Kumar
If 3 sin θ + 5 cos θ = 5, find the value of 5 sin θ − 3 cos θ.
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Answer: C
Solution: Square and add the two expressions: \((3\sin\theta+5\cos\theta)^2+(5\sin\theta-3\cos\theta)^2=34(\sin^2\theta+\cos^2\theta)=34\). The first squared expression equals \(5^2=25\). Therefore, \((5\sin\theta-3\cos\theta)^2=34-25=9\), so \(5\sin\theta-3\cos\theta=\pm3\). Trick: Choose coefficients so that the cross terms cancel when the expressions are squared and added.
यदि 3 sin θ + 5 cos θ = 5 है, तो 5 sin θ − 3 cos θ का मान ज्ञात कीजिए।
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±1
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उत्तर: C
हल: दोनों व्यंजकों के वर्ग करके जोड़ें: \((3\sin\theta+5\cos\theta)^2+(5\sin\theta-3\cos\theta)^2=34(\sin^2\theta+\cos^2\theta)=34\)। पहले वर्ग का मान \(5^2=25\) है। इसलिए \((5\sin\theta-3\cos\theta)^2=34-25=9\), अतः \(5\sin\theta-3\cos\theta=\pm3\)। ट्रिक: गुणांक इस प्रकार चुनें कि दोनों व्यंजकों के वर्गों को जोड़ने पर मिश्रित पद कट जाएँ।
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