Question

For any real number \(x\), the least value of \(4cosx-3sinx+5\) is ?

• A. \(10\)
• B. \(2\)
• C. \(0\)
• D. \(8\)
• E. \(4\)

Answer 1

The Correct Option is C

Given:

• The expression \(4\cos{x} - 3\sin{x} + 5\) for any real number \(x\)

Step 1: Express the trigonometric combination in amplitude-phase form: \[ 4\cos{x} - 3\sin{x} = 5\cos(x + \alpha) \] where \( \alpha \) is an angle satisfying \(\cos{\alpha} = \frac{4}{5}\) and \(\sin{\alpha} = \frac{3}{5}\), and the amplitude \(R = \sqrt{4^2 + (-3)^2} = 5\).

Step 2: Determine the range of the expression: \[ -5 \leq 5\cos(x + \alpha) \leq 5 \] Adding 5: \[ 0 \leq 5\cos(x + \alpha) + 5 \leq 10 \]

Step 3: Identify the least value: The minimum value occurs when \(\cos(x + \alpha) = -1\): \[ 4\cos{x} - 3\sin{x} + 5 = 5(-1) + 5 = 0 \]

The least value of the expression is (C) 0.

Answer 2

Given

\(4cosx-3sinx+5\)

for which we need to find the least value for any real number \(x\).

So we can proceed by using Pythagoras identity for cosine i.e. 

\(cos²(x) + sin²(x) = 1\)

Now, let's find the angle whose cosine is 4/5 and sine is 3/5. Let θ be that angle:

\(cos(θ) = 4/5\) , \(sin(θ) = 3/5\)

Using the Pythagorean identity:

\(cos²(θ) + sin²(θ) = 1\)

\((4/5)² + (3/5)² = 1\)

\(16/25 + 9/25 = 1\)

\(25/25 = 1\)

Now, we have θ such that \(cos(θ) = 4/5\) and \(sin(θ) = 3/5\). 

So, \(4cos(x) - 3sin(x)\) can be rewritten as:

\(4cos(x) - 3sin(x) = 4cos(θ) - 3sin(θ)\)

Using the angle sum identity for cosine:

\(4cos(θ) - 3sin(θ) = 5cos(θ + α)\)

where α is the angle such that :

\(cos(α) = 3/5\) and \(sin(α) = 4/5\)

By again using Pythagorean identity we can write:

\(cos²(α) + sin²(α) = 1\)

\((3/5)² + (4/5)² = 1\)

\(9/25 + 16/25 = 1\)

\(25/25 = 1\)

So, we have α such that \(cos(α) = 3/5\) and \(sin(α) = 4/5\).

Now,\(4cos(x) - 3sin(x) = 5cos(θ + α)\)

The least value of \(5cos(θ + α)\) is \(-5\), at \(cos(θ + α) = -1\).

Therefore, the least value of \(4cos(x) - 3sin(x) + 5\) is:

\(4cos(x) - 3sin(x) + 5 = 5cos(θ + α) + 5 = -5 + 5 = 0\)

So, the least value of the expression is \(0\)