Minimum value of (3 sin$^2$ θ + 4 cos$^2$ θ) is :

Q: Minimum value of (3 sin$^2$ θ + 4 cos$^2$ θ) is :

To determine the minimum value of the expression $3 \sin^2 \theta + 4 \cos^2 \theta$, we need to analyze it using trigonometric identities and algebraic manipulation. ### Explanation: 1. **Expression Analysis:** The given expression is $3 \sin^2 \theta + 4 \cos^2 \theta$. We know from the Pythagorean identity that $\sin^2 \theta + \cos^2 \theta = 1$. This allows us to express one trigonometric function in terms of the other. 2. **Substitution:** We can express $\sin^2 \theta$ as $1 - \cos^2 \theta$. Substituting this into the expression gives: \[ 3(1 - \cos^2 \theta) + 4 \cos^2 \theta = 3 - 3 \cos^2 \theta + 4 \cos^2 \theta \] Simplifying this, we get: \[ 3 + \cos^2 \theta \] 3. **Finding the Minimum Value:** The expression $3 + \cos^2 \theta$ depends on $\cos^2 \theta$, which varies between 0 and 1 (since $\cos^2 \theta$ is the square of the cosine function, which ranges from -1 to 1). The minimum value of $\cos^2 \theta$ is 0. 4. **Calculating the Minimum:** When $\cos^2 \theta = 0$, the expression becomes: \[ 3 + 0 = 3 \] Thus, the minimum value of the expression $3 \sin^2 \theta + 4 \cos^2 \theta$ is indeed 3, which corresponds to Option 4. ### Why Other Options are Incorrect: - **Option 1 (0):** This is incorrect because the expression $3 + \cos^2 \theta$ cannot be less than 3, as $\cos^2 \theta$ is always non-negative. - **Option 2 (1):** This is incorrect because the minimum value of $3 + \cos^2 \theta$ is 3, not 1. The value of $\cos^2 \theta$ cannot make the expression equal to 1. - **Option 3 (2):** This is incorrect because, similarly, the minimum value of the expression is 3, not 2. The expression $3 + \cos^2 \theta$ cannot be reduced to 2 given the range of $\cos^2 \theta$. Therefore, the correct answer is Option 4, which is 3.

Question

Accepted Answer

To determine the minimum value of the expression $3 \sin^2 \theta + 4 \cos^2 \theta$, we need to analyze it using trigonometric identities and algebraic manipulation.

### Explanation:

1. **Expression Analysis:**
   The given expression is $3 \sin^2 \theta + 4 \cos^2 \theta$. We know from the Pythagorean identity that $\sin^2 \theta + \cos^2 \theta = 1$. This allows us to express one trigonometric function in terms of the other.

2. **Substitution:**
   We can express $\sin^2 \theta$ as $1 - \cos^2 \theta$. Substituting this into the expression gives:
   $$
   3(1 - \cos^2 \theta) + 4 \cos^2 \theta = 3 - 3 \cos^2 \theta + 4 \cos^2 \theta
   $$
   Simplifying this, we get:
   $$
   3 + \cos^2 \theta
   $$

3. **Finding the Minimum Value:**
   The expression $3 + \cos^2 \theta$ depends on $\cos^2 \theta$, which varies between 0 and 1 (since $\cos^2 \theta$ is the square of the cosine function, which ranges from -1 to 1). The minimum value of $\cos^2 \theta$ is 0.

4. **Calculating the Minimum:**
   When $\cos^2 \theta = 0$, the expression becomes:
   $$
   3 + 0 = 3
   $$

Thus, the minimum value of the expression $3 \sin^2 \theta + 4 \cos^2 \theta$ is indeed 3, which corresponds to Option 4.

### Why Other Options are Incorrect:

- **Option 1 (0):** This is incorrect because the expression $3 + \cos^2 \theta$ cannot be less than 3, as $\cos^2 \theta$ is always non-negative.

- **Option 2 (1):** This is incorrect because the minimum value of $3 + \cos^2 \theta$ is 3, not 1. The value of $\cos^2 \theta$ cannot make the expression equal to 1.

- **Option 3 (2):** This is incorrect because, similarly, the minimum value of the expression is 3, not 2. The expression $3 + \cos^2 \theta$ cannot be reduced to 2 given the range of $\cos^2 \theta$.

Therefore, the correct answer is Option 4, which is 3.