Q. 173.5( 4 Votes )

# If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.

Answer :

⇒ 3 sin x +5cos x = 5

⇒ 3sin x = 5-5cos x

⇒ 3sin x = 5(1-cos x)

Squaring both sides we get

⇒ 9sin^{2}x = 25(1-cos x)^{2}

⇒ 9sin^{2}x = 25(1+cos^{2}x-2cos x)

⇒ 9sin^{2}x+9cos^{2}x = 25 + 25cos^{2}x - 50cos x + 9cos^{2}x

⇒ 9(sin^{2}x+ cos^{2}x) = 25 +34cos^{2}x-50cos x

⇒ 34cos^{2}x-50cos x+16=0

⇒ 17cos^{2}x-25cos x+8=0

⇒ 17cos^{2}x-17cos x-8cos x+8=0

⇒ 17cos x(cos x-1)-8(cos x-1)=0

When cos x = 1

3sin x + 5cos x = 5

3sin x = 0

Sin x = 0

Substituting the value cos x = 1 and sin x = 0

5(0)-3(1) = 0-3

⇒ -3.

⇒

⇒ sin^{2}x+ cos^{2}x = 1

⇒ sin^{2}x = 1- cos^{2}x

⇒ 5sin x – 3cos x

∴ -3 and 3.

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