Question

The value of \( 5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) + 3 \) lies between:

• A. -2 and 5
• B. -1 and 8
• C. -3 and 6
• D. -4 and 10

Hint:

Use trigonometric identities to simplify expressions and find the range of the trigonometric function.

Answer 1

The Correct Option is D

We are given the expression: \( 5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) + 3. \)
To simplify, we will use the sum identity for cosine: \( \cos \left( \theta + \frac{\pi}{3} \right) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3}. \)
Since \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), we substitute these values into the expression: \( \cos \left( \theta + \frac{\pi}{3} \right) = \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta. \)
Now, substitute this into the original expression: \( 5 \cos \theta + 3 \left( \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \right) + 3. \)
Simplifying: \( 5 \cos \theta + \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3. \)
Combine like terms: \( \left( 5 + \frac{3}{2} \right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3. \)
This simplifies to: \( \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3. \)
Now, we need to find the range of this expression. It is a linear combination of sine and cosine functions, which can be written in the form \( R \cos (\theta - \alpha) \), where \( R \) is the resultant amplitude and \( \alpha \) is the phase shift. The amplitude \( R \) is given by:
\( R = \sqrt{\left( \frac{13}{2} \right)^2 + \left( \frac{3\sqrt{3}}{2} \right)^2} = \sqrt{\frac{169}{4} + \frac{27}{4}} = \sqrt{\frac{196}{4}} = \sqrt{49} = 7. \)
Thus, the maximum value of \( \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \) is 7, and the minimum value is -7. Now, adding the constant term 3:
\( \text{Maximum value} = 7 + 3 = 10, \)
\( \text{Minimum value} = -7 + 3 = -4. \) Therefore, the value of the expression lies between \( -4 \) and \( 10 \). Thus, the correct answer is: \( \boxed{(D) \, -4 \, \text{and} \, 10}. \)