Find the minimum value of the expression 4cos(x)-3sin(x)-4.
Give the smallest possible positive value of x for which this minimum value occurs.
Show all steps in Edexcel Alevel Maths form

We are tasked with finding the minimum value of the expression \( 4\cos(x) - 3\sin(x) - 4 \), and determining the smallest positive value of \( x \) for which this minimum value occurs.

### Step 1: Rewrite the expression in a simplified form

We start by writing \( 4\cos(x) - 3\sin(x) \) as a single trigonometric function. The general form of such an expression can be written as:

\[
R\cos(x + \alpha)
\]

where \( R \) and \( \alpha \) are constants. To find these constants, we use the identities:

\[
R\cos(x + \alpha) = R\cos(x)\cos(\alpha) - R\sin(x)\sin(\alpha)
\]

By comparing coefficients with the given expression \( 4\cos(x) - 3\sin(x) \), we have:

\[
R\cos(\alpha) = 4 \quad \text{and} \quad R\sin(\alpha) = -3
\]

### Step 2: Solve for \( R \) and \( \alpha \)

To find \( R \), we use the Pythagorean identity:

\[
R = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

Next, we find \( \alpha \) using the tangent function:

\[
\tan(\alpha) = \frac{-3}{4}
\]

Thus,

\[
\alpha = \tan^{-1}\left(\frac{-3}{4}\right)
\]

This value for \( \alpha \) is in the fourth quadrant (since both cosine and sine are involved). We will leave \( \alpha \) as \( \tan^{-1}\left(\frac{-3}{4}\right) \) for now, but it’s essential to understand that it is a negative angle.

### Step 3: Rewrite the original expression

The original expression \( 4\cos(x) - 3\sin(x) - 4 \) can now be rewritten as:

\[
5\cos(x + \alpha) - 4
\]

### Step 4: Find the minimum value

The function \( 5\cos(x + \alpha) \) oscillates between \( -5 \) and \( 5 \), so the minimum value of \( 5\cos(x + \alpha) \) is \( -5 \).

Substituting this into the expression \( 5\cos(x + \alpha) - 4 \), we get:

\[
\text{Minimum value} = -5 - 4 = -9
\]

Thus, the minimum value of the expression is \( -9 \).

### Step 5: Find the smallest positive value of \( x \)

The minimum of \( \cos(x + \alpha) \) occurs when \( x + \alpha = \pi \), since \( \cos(\pi) = -1 \).

So,

\[
x + \alpha = \pi
\]

Solving for \( x \), we get:

\[
x = \pi - \alpha = \pi - \tan^{-1}\left(\frac{-3}{4}\right)
\]

Since \( \alpha \) is negative, \( \pi - \alpha \) gives a positive value. Therefore, the smallest positive value of \( x \) is:

\[
x = \pi - \alpha
\]

### Final Answer:
- The minimum value of the expression is \( -9 \).
- The smallest positive value of \( x \) is \( \pi - \alpha \), where \( \alpha = \tan^{-1}\left(\frac{-3}{4}\right) \).

Would you like further details or clarifications on any step?

Here are some related questions for practice:
1. What is the maximum value of \( 4\cos(x) - 3\sin(x) - 4 \)?
2. How would the result change if the expression was \( 4\sin(x) - 3\cos(x) - 4 \)?
3. Find the range of the expression \( 4\cos(x) - 3\sin(x) - 4 \).
4. What happens if you replace 4 and 3 with other constants?
5. How does the method change if we are looking for the maximum instead of the minimum?

**Tip:** When rewriting trigonometric expressions as a single cosine or sine, always verify the quadrant of the angle using the signs of the coefficients.

Find the minimum value of the expression 4cos(x)-3sin(x)-4. Give the smallest possible positive value of x for which this minimum value occurs. Show all steps in Edexcel Alevel Maths form.

Learn how to find the minimum value of 4cos(x)-3sin(x)-4 and determine the smallest positive x where this minimum occurs. This step-by-step solution involves trigonometric function transformation and key formulas like Rcos(x+α). Ideal for A-level Maths students.

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