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Question
If a cos θ − b sin θ = c, show that a sin θ + b cos θ = `+- sqrt("a"^2 + "b"^2 - "c"^2)`
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Solution
a cos θ – b sin θ = c
(a cos θ – b sin θ)2 + (a sin θ + b cos θ)2 = a2 cos2 θ – 2 ab sin θ cos θ + b2 sin2θ + a2 sin2 θ + b2 cos2 θ + 2 ab sin θ cos θ
c2 + (a sin 0 + b cos θ )2 = a2 cos2 θ + a2 sin2 θ + b2 sin2 θ + b2cos2θ
= a2 (cos2θ + sin2θ) + b2(sin2θ + cos2θ)
c2 + (a sin θ + b cos θ)2 = a2 + b2
(a sin θ + b cos θ)2 = a2 + b2 – c2
a sin θ + b cos θ = `+- sqrt("a"^2 + "b"^2 - "c"^2)`
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