Question

If $3\sin \theta +5\cos \theta =5$, then the number of integer value(s) of expression $(5\sin \theta -3\cos \theta )$ takes place is (are) :

• A. $3$
• B. $4$
• C. $5$
• D. $9$

Answer 1

The correct option is A $3$

Given,

$3sin\theta +5cos\theta =5$

Now, squaring both sides, we get

$(3sin\theta +5cos\theta )²=5²$

$9sin²\theta +30sin\theta cos\theta +25cos²\theta =25$

$9(1-cos²\theta )+30sin\theta cos\theta +25(1-sin²\theta )=25$

$9-9cos²\theta +30sin\theta cos\theta +25-25sin²\theta =25$

$25sin²\theta -30sin\theta cos\theta +9cos²\theta =9$

$(5sin\theta -3cos\theta )²=3²$

$5sin\theta -3cos\theta =3,$

which is the required answer