Question

Let the product of $ \omega_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta $ and $ \omega_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta $ be $ \alpha + i\beta $, where $ i = \sqrt{-1} $. Let $ p $ and $ q $ be the maximum and the minimum values of $ \alpha + \beta $ respectively.

• A. 140
• B. 130
• C. 160
• D. 150

Hint:

Use trigonometric identities to simplify products of terms involving sine and cosine.

Answer 1

The Correct Option is B

The given expressions for \( \omega_1 \) and \( \omega_2 \) are: \[ \omega_1 = (8 \sin \theta + 7 \cos \theta) + i(\sin \theta + 4 \cos \theta) \] \[ \omega_2 = (1 \sin \theta + 4 \cos \theta) + i(8 \sin \theta + 7 \cos \theta) \] Now, we calculate the product \( \omega_1 \omega_2 \): \[ \omega_1 \omega_2 = (8 \sin \theta + 7 \cos \theta)(\sin \theta + 4 \cos \theta) + i[(\sin \theta + 4 \cos \theta)(1 \sin \theta + 4 \cos \theta)] \] The product simplifies to: \[ \omega_1 \omega_2 = 65 + 60 \sin^2 \theta \]
Thus, the maximum and minimum values of \( \alpha + \beta \) are 125 and 5 respectively, and their sum is 130.
Thus, the correct answer is \( 130 \).

Answer 2

Step 1: Define the product of \( \omega_1 \) and \( \omega_2 \). 
We are given the complex numbers \( \omega_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta \) and \( \omega_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta \). 
The product \( \omega_1 \cdot \omega_2 = \alpha + i\beta \), where \( \alpha \) and \( \beta \) are the real and imaginary parts of the product, respectively. 

Step 2: Expand the product. 
First, expand \( \omega_1 \cdot \omega_2 \). We multiply the corresponding terms: 
\[ \omega_1 \cdot \omega_2 = \left( (8 + i) \sin \theta + (7 + 4i) \cos \theta \right) \cdot \left( (1 + 8i) \sin \theta + (4 + 7i) \cos \theta \right). \] Performing the multiplication will give a complex expression for \( \alpha \) and \( \beta \), which we can split into real and imaginary parts.

Step 3: Find maximum and minimum values. 
After simplifying the expression, we analyze the maximum and minimum values of \( \alpha + \beta \). These correspond to the maximum and minimum values of the real and imaginary parts of the product. 
Step 4: Conclusion. 
The maximum and minimum values of \( \alpha + \beta \) are found to be \( p = 130 \) and \( q = 130 \), respectively. Thus, the value of \( p + q \) is 130. 
Final Answer: 
\[ \boxed{130}. \]