Q3. The value of 3 cos θ + 4 sin θ lies in :

(a) [-5, 5]
(b) (-5, 5)
(c) (-5, 5]
(d) None of these​

2 thoughts on “Q3. The value of 3 cos θ + 4 sin θ lies in :<br /><br />(a) [-5, 5]<br />(b) (-5, 5)<br />(c) (-5, 5]<br />(d) None of these​”

1. ANSWER:

   To Find:

   • Value of 3 cos θ + 4 sin θ lies in?

   Solution:

   To find where the value of 3 cos θ + 4 sin θ lies in, we will find the maximum and minimum values of the expression.

   So,

   We know that, Maximum Value of (a cos θ + b sin θ) is,

   ⇒ √(a²+b²)

   Here, a = 3 and b = 4. So,

   Maximum Value of 3 cos θ + 4 sin θ

   ⇒ √(3²+4²) ⇒ √(9+16) ⇒ √(25) ⇒ +5 —–(1)

   And,

   We know that, Minimum Value of (a cos θ + b sin θ) is,

   ⇒ -√(a²+b²)

   Here, a = 3 and b = 4. So,

   Minimum Value of 3 cos θ + 4 sin θ

   ⇒ -√(3²+4²) ⇒ -√(9+16) ⇒ -√(25) ⇒ -5 —–(2)

   So, the value of 3 cos θ + 4 sin θ lies in,

   ⇒ [Minimum Value , Maximum Value]

   (We are using square brackets[] because, both the maximum and minimum values are inclusive.)

   From (1) & (2),

   Hence, the value of 3 cos θ + 4 sin θ lies in,

   ⇒ [-5 , 5]

   (a) [-5, 5] is the answer.

   Formula Used:

   • Maximum Value of (a cos θ + b sin θ) is, √(a²+b²)
   • Minimum Value of (a cos θ + b sin θ) is, -√(a²+b²)

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2. Step-by-step explanation:

   We know that the maximum value of a cos θ + b sin θ is √(a2 + b2).

   Substituting a = 3, b = 4,

   √(a2 + b2) = √(9 + 16) = √25 = 5

   Therefore, the maximum value of 3 cos θ + 4 sin θ is 5.

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