Question

The general solution of \(cos4θ = cos3θ\) is \(θ =\).....

• A. \(\frac{2nπ}{7},n\in Z\)
• B. \(\frac{2nπ}{5},n\in Z\)
• C. \(nπ,n\in Z\)
• D. \(\frac{4nπ}{3},n\in Z\)

Hint:

Use cos A = cos B gives A = 2n pi plus or minus B.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
\(\cos A=\cos B\) means \(A=2n\pi\pm B\), \(n\in Z\).

Step 2: Apply with A = 4theta, B = 3theta:
Case 1: \(4\theta=2n\pi+3\theta\), so \(\theta=2n\pi\). Case 2: \(4\theta=2n\pi-3\theta\), so \(7\theta=2n\pi\) and \(\theta=\dfrac{2n\pi}{7}\).

Step 3: Combine:
Case 1 values (\(\theta=2n\pi\)) are already contained in case 2 when \(n\) is a multiple of 7. So the full solution is \(\theta=\dfrac{2n\pi}{7}\), \(n\in Z\).

Step 4: Why the other options are wrong.
\(2n\pi/5\), \(n\pi\) and \(4n\pi/3\) do not satisfy the equation. For example \(\theta=\pi\) gives \(\cos4\pi=1\) but \(\cos3\pi=-1\).

Final Answer:
The general solution is 2n pi / 7. \[ \boxed{\text{(A) }\dfrac{2n\pi}{7},\ n\in Z} \]