Question

If \(cos(pθ)+cos(qθ) = 0\) and \(p\neq q\), then the general value of \(θ\) (where n is any integer) is...

• A. \(\frac{(3n+1)π}{p-q}\)
• B. \(\frac{(2n+1)π}{p\pm q}\)
• C. \(\frac{(n\pm 1)π}{p\pm q}\)
• D. \(\frac{(n+2)π}{p+q}\)

Hint:

Use the sum-to-product formula and set each cosine to zero.

Answer 1

The Correct Option is B

Step 1: Sum to product:
$\cos p\theta+\cos q\theta=2\cos\frac{(p+q)\theta}{2}\cos\frac{(p-q)\theta}{2}=0$.

Step 2: Each factor:
Either $\cos\frac{(p+q)\theta}{2}=0$ giving $\frac{(p+q)\theta}{2}=(2n+1)\frac\pi2$, so $\theta=\frac{(2n+1)\pi}{p+q}$.

Step 3: Other factor:
Or $\cos\frac{(p-q)\theta}{2}=0$ giving $\theta=\frac{(2n+1)\pi}{p-q}$.

Step 4: Match:
Together, $\theta=\frac{(2n+1)\pi}{p\pm q}$. Option (B).

Final Answer:
Product form splits into two cosine zero conditions. \[ \boxed{B} \]