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Minimum value of (1)/(3 sin theta - 4 co...

Minimum value of `(1)/(3 sin theta - 4 cos theta + 7)` is

A

`(7)/(12)`

B

`(5)/(12)`

C

`(1)/(12)`

D

`(1)/(6)`

Text Solution

AI Generated Solution

The correct Answer is:
To find the minimum value of the expression \( \frac{1}{3 \sin \theta - 4 \cos \theta + 7} \), we will first determine the maximum value of the expression in the denominator, \( 3 \sin \theta - 4 \cos \theta + 7 \). ### Step 1: Find the maximum value of \( 3 \sin \theta - 4 \cos \theta \) To find the maximum value of the expression \( 3 \sin \theta - 4 \cos \theta \), we can rewrite it in the form \( R \sin(\theta + \phi) \), where \( R = \sqrt{a^2 + b^2} \) and \( a \) and \( b \) are the coefficients of \( \sin \theta \) and \( \cos \theta \) respectively. Here, \( a = 3 \) and \( b = -4 \). \[ R = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] ### Step 2: Determine the angle \( \phi \) Next, we calculate the angle \( \phi \) using the relationships: \[ \sin \phi = \frac{-4}{5} \quad \text{and} \quad \cos \phi = \frac{3}{5} \] ### Step 3: Rewrite the expression Now we can express \( 3 \sin \theta - 4 \cos \theta \) as: \[ 3 \sin \theta - 4 \cos \theta = 5 \sin(\theta + \phi) \] ### Step 4: Find the maximum value The maximum value of \( \sin \) function is 1. Therefore, the maximum value of \( 5 \sin(\theta + \phi) \) is: \[ 5 \cdot 1 = 5 \] ### Step 5: Add 7 to find the maximum of the entire expression Now we can find the maximum value of \( 3 \sin \theta - 4 \cos \theta + 7 \): \[ \text{Maximum of } (3 \sin \theta - 4 \cos \theta + 7) = 5 + 7 = 12 \] ### Step 6: Find the minimum value of the original expression Now that we have the maximum value of the denominator, we can find the minimum value of the original expression: \[ \text{Minimum value of } \frac{1}{3 \sin \theta - 4 \cos \theta + 7} = \frac{1}{12} \] Thus, the minimum value of \( \frac{1}{3 \sin \theta - 4 \cos \theta + 7} \) is: \[ \boxed{\frac{1}{12}} \]
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Knowledge Check

  • The minimum value of 3 sin theta + 4 cos theta is

    A
    1
    B
    3
    C
    `-5`
    D
    5
  • The minimum value of 3 sin theta+ 4 cos theta is:

    A
    5
    B
    1
    C
    3
    D
    -5
  • The minimum value of sin^2 theta + cos^4 theta is

    A
    `1/sqrt2`
    B
    `3/5`
    C
    `3/4`
    D
    `2/3`
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