Q. 145.0( 2 Votes )

# If (a cos θ - b sin θ) = c, show that: (a sin θ + b cos θ) = .

Given: (a cos θ - b sin θ) = c ……………… (1)
Consider: (a cos θ - b sin θ) 2 + (a sin θ + b cos θ) 2
= a 2 cos 2 θ + b 2 sin 2 θ – 2ab sin θ cos θ + a 2 sin 2 θ + b 2 cos 2 θ + 2ab sin θ cos θ

= a 2 (cos 2 θ + sin 2 θ ) + b 2 (sin 2 θ + cos 2 θ ) – 2ab sin θ cos θ + 2ab sin θ cos θ
= a 2 + b 2 (
sin 2 θ + cos 2 θ = 1)

(a cos θ - b sin θ) 2 + (a sin θ + b cos θ) 2 = a 2 + b 2
(a cos θ - b sin θ) 2 = a 2 + b 2 - (a sin θ + b cos θ) 2
= a 2 + b 2 - c 2 (from equation (1))

(a sin θ + b cos θ) = .
Hence, proved.

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