if 3sinA+5cosA=5, prove that 3sinA+5cosA=plus or minus 3

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Solution

3sinA + 5cosA = 5

on squaring both side

we get,

9sin²A + 25cos²A + 30sinAcosA = 25

= 9(1 - cos²A) + 25(1 - sin²A) + 30sinAcosA = 25

= 9 - 9cos²A + 25 - 25sin²A + 30sinAcosA = 25

= 9 + 25 - 25 = 9cos²A + 25sin²A - 30sinAcosA

= (3cosA) + (5sinA) - 2×3×5sinAcosA = 9

= (3cosA - 5sinA)² = 9

= 3cosA - 5sinA = +-3

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