# 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​

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​

### 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)