Minimum value of `5costheta + 12` is:

Options

A. 5
B. 12
C. 7
D. 17

Answer

To find the minimum value of the expression \(5 \cos \theta + 12\), we can follow these steps:

### Step 1: Understand the range of \(\cos \theta\)
The cosine function, \(\cos \theta\), ranges from \(-1\) to \(1\). This means that the minimum value of \(\cos \theta\) is \(-1\) and the maximum value is \(1\).

**Hint:** Remember the basic properties of the cosine function and its range.

### Step 2: Substitute the minimum value of \(\cos \theta\)
To find the minimum value of the expression \(5 \cos \theta + 12\), we substitute the minimum value of \(\cos \theta\) into the expression. The minimum value of \(\cos \theta\) is \(-1\).

\[ 5 \cos \theta + 12 = 5(-1) + 12 \]

**Hint:** When looking for minimum values, always substitute the extreme values of the function involved.

### Step 3: Calculate the expression
Now, we calculate:

\[ 5(-1) + 12 = -5 + 12 \]

**Hint:** Perform the arithmetic carefully to avoid mistakes.

### Step 4: Simplify the result
Now, simplify the expression:

\[ -5 + 12 = 7 \]

**Hint:** Double-check your arithmetic to ensure accuracy.

### Conclusion
The minimum value of \(5 \cos \theta + 12\) is \(7\).

**Final Answer:** The minimum value is \(7\).