Trigonometry & Inverse Trigonometry
General Solution of Trigonometric Equations
Grade 11
Question:
<p>Consider <i>f</i>, <i>g</i> and <i>h</i> be three real valued functions defined on ℝ.<br/>Let <i>f</i>(<i>x</i>) = sin 3<i>x</i> + cos <i>x</i>, <i>g</i>(<i>x</i>) = cos 3<i>x</i> + sin <i>x</i> and <i>h</i>(<i>x</i>) = <i>f</i>²(<i>x</i>) + <i>g</i>²(<i>x</i>)<br/><br/>General solution of the equation <i>h</i>(<i>x</i>) = 4, is:</p>
<p>(a) \((4n + 1)\frac{\pi}{8}\)</p>
<p>(b) \((8n + 1)\frac{\pi}{8}\)</p>
<p>(c) \((2n + 1)\frac{\pi}{4}\)</p>
<p>(d) \((7n + 1)\frac{\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Substitute the simplified form h(x) = 2 + 2sin(4x), set equal to 4, and solve the resulting trigonometric equation.
<p><strong>Solution:</strong> From the previous part, \(h(x) = 2 + 2\sin 4x\)</p><p>Setting \(h(x) = 4\):</p><p>\(2 + 2\sin 4x = 4\)</p><p>\(\sin 4x = 1\)</p><p>\(4x = \frac{\pi}{2} + 2\pi k\) where \(k \in \mathbb{Z}\)</p><p>\(x = \frac{\pi}{8} + \frac{\pi k}{2} = (4n + 1)\frac{\pi}{8}\) where \(n \in \mathbb{Z}\)</p><p>∴ Answer is (a)</p>
Correct Answer: A