Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If \(0 \leq x < 2\pi\), then the number of real values of \(x\), which satisfy the equation \(\cos x + \cos 2x + \cos 3x + \cos 4x = 0\), is</p>
<p>(a) 3</p>
<p>(b) 5</p>
<p>(c) 7</p>
<p>(d) 9</p>
Step-by-Step Solution
Key Concept: Group cosine terms strategically and apply sum-to-product formulas to factor the expression.
<p>Group the cosines: \((\cos x + \cos 4x) + (\cos 2x + \cos 3x) = 0\). Using sum-to-product formulas: \(2\cos\frac{5x}{2}\cos\frac{3x}{2} + 2\cos\frac{5x}{2}\cos\frac{x}{2} = 0\), which factors to give 7 solutions in \([0, 2\pi)\).</p>
Correct Answer: c