Key Concepts and Formulas

• Product of Complex Numbers: If $z_1 = a+ib$ and $z_2 = c+id$, then $z_1 z_2 = (ac-bd) + i(ad+bc)$.
• Trigonometric Identities:
  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $2 \sin \theta \cos \theta = \sin 2\theta$
• Range of Sine Function: $-1 \le \sin x \le 1$.

Step-by-Step Solution

1. Express $\omega_1$ and $\omega_2$ in the form $a+ib$

We rearrange the given complex numbers $\omega_1$ and $\omega_2$ to separate their real and imaginary parts. This facilitates their multiplication.

Given: $\omega_1 = (8+i) \sin \theta+(7+4 i) \cos \theta$ Distribute $\sin \theta$ and $\cos \theta$: $\omega_1 = 8 \sin \theta + i \sin \theta + 7 \cos \theta + 4i \cos \theta$ Group the real and imaginary terms: $\omega_1 = (8 \sin \theta + 7 \cos \theta) + i(\sin \theta + 4 \cos \theta)$ Let $A = 8 \sin \theta + 7 \cos \theta$ and $B = \sin \theta + 4 \cos \theta$. Then, $\omega_1 = A + iB$.

Similarly for $\omega_2$: $\omega_2 = (1+8 i) \sin \theta+(4+7 i) \cos \theta$ Distribute $\sin \theta$ and $\cos \theta$: $\omega_2 = \sin \theta + 8i \sin \theta + 4 \cos \theta + 7i \cos \theta$ Group the real and imaginary terms: $\omega_2 = (\sin \theta + 4 \cos \theta) + i(8 \sin \theta + 7 \cos \theta)$ The real part of $\omega_2$ is $B$ and the imaginary part is $A$. Thus, $\omega_2 = B + iA$.

2. Calculate the Product $\omega_1 \omega_2$

We find the product $\omega_1 \omega_2 = (A+iB)(B+iA)$.

$\omega_1 \omega_2 = (A+iB)(B+iA)$ Expand the product: $\omega_1 \omega_2 = AB + iA^2 + iB^2 + i^2 AB$ Since $i^2 = -1$: $\omega_1 \omega_2 = AB + iA^2 + iB^2 - AB$ $\omega_1 \omega_2 = i(A^2 + B^2)$ We are given that the product is $\alpha + i\beta$. Comparing real and imaginary parts: $\alpha + i\beta = 0 + i(A^2 + B^2)$ Therefore, $\alpha = 0$ and $\beta = A^2 + B^2$.

3. Simplify the Expression for $\alpha + \beta$

Since $\alpha = 0$, we have $\alpha + \beta = \beta = A^2 + B^2$. Substitute the expressions for $A$ and $B$: $A^2 = (8 \sin \theta + 7 \cos \theta)^2$ $B^2 = (\sin \theta + 4 \cos \theta)^2$ Expand $A^2$: $A^2 = 64 \sin^2 \theta + 49 \cos^2 \theta + 2(8 \sin \theta)(7 \cos \theta)$ $A^2 = 64 \sin^2 \theta + 49 \cos^2 \theta + 112 \sin \theta \cos \theta$ Expand $B^2$: $B^2 = \sin^2 \theta + 16 \cos^2 \theta + 2(\sin \theta)(4 \cos \theta)$ $B^2 = \sin^2 \theta + 16 \cos^2 \theta + 8 \sin \theta \cos \theta$ Add $A^2$ and $B^2$: $A^2 + B^2 = (64 \sin^2 \theta + 49 \cos^2 \theta + 112 \sin \theta \cos \theta) + (\sin^2 \theta + 16 \cos^2 \theta + 8 \sin \theta \cos \theta)$ Group like terms: $A^2 + B^2 = (64+1) \sin^2 \theta + (49+16) \cos^2 \theta + (112+8) \sin \theta \cos \theta$ $A^2 + B^2 = 65 \sin^2 \theta + 65 \cos^2 \theta + 120 \sin \theta \cos \theta$ Factor out $65$: $A^2 + B^2 = 65 (\sin^2 \theta + \cos^2 \theta) + 120 \sin \theta \cos \theta$ Apply the identity $\sin^2 \theta + \cos^2 \theta = 1$: $A^2 + B^2 = 65(1) + 120 \sin \theta \cos \theta$ Apply the identity $2 \sin \theta \cos \theta = \sin 2\theta$: $A^2 + B^2 = 65 + 60 (2 \sin \theta \cos \theta)$ $A^2 + B^2 = 65 + 60 \sin 2\theta$ So, $\alpha + \beta = 65 + 60 \sin 2\theta$.

4. Determine the Maximum (p) and Minimum (q) Values of $\alpha + \beta$

We have $\alpha + \beta = 65 + 60 \sin 2\theta$. The range of the sine function is $[-1, 1]$. This means $-1 \le \sin 2\theta \le 1$.

To find the maximum value, we use $\sin 2\theta = 1$: $p = (\alpha + \beta)_{\max} = 65 + 60(1) = 65 + 60 = 125$

To find the minimum value, we use $\sin 2\theta = -1$: $q = (\alpha + \beta)_{\min} = 65 + 60(-1) = 65 - 60 = 5$

5. Calculate $p+q$

Finally, we sum the maximum and minimum values found: $p+q = 125 + 5 = 130$

Common Mistakes & Tips

• Ensure accurate distribution of trigonometric terms when separating real and imaginary parts of complex numbers.
• Remember the correct formula for multiplying complex numbers: $(a+ib)(c+id) = (ac-bd) + i(ad+bc)$.
• Be fluent with trigonometric identities like $\sin^2\theta + \cos^2\theta = 1$ and $2\sin\theta\cos\theta = \sin 2\theta$.

Summary

This problem elegantly combines complex number manipulation with trigonometric identities. By expressing the complex numbers in standard form, performing multiplication, simplifying using trigonometric identities, and leveraging the range of the sine function, we determined that $p+q = 130$.

Final Answer

The final answer is $\boxed{130}$, which corresponds to option (A).