Trigonometry & Inverse Trigonometry
Multiple Angle Formulas
Grade 11
Question:
<p>If <span class="math">\cos 5x = a \cos^5 x + b \cos^3 x + c \cos x + d</span>, then</p>
<p>(a) <span class="math">a = 16</span></p>
<p>(b) <span class="math">b = -20</span></p>
<p>(c) <span class="math">c = 5</span></p>
<p>(d) <span class="math">d = 2</span></p>
Step-by-Step Solution
Key Concept: Use Chebyshev polynomials to express multiple angles as polynomial functions of the base trigonometric function.
<p>Using the multiple angle formula for <span class="math">\cos 5x</span>, we can derive it in terms of <span class="math">\cos x</span> using Chebyshev polynomials.</p><p>The Chebyshev polynomial <span class="math">T_5(\cos x) = \cos 5x</span> is given by:</p><p><span class="math">T_5(t) = 16t^5 - 20t^3 + 5t</span></p><p>Therefore, <span class="math">\cos 5x = 16\cos^5 x - 20\cos^3 x + 5\cos x</span>.</p><p>Comparing with <span class="math">a \cos^5 x + b \cos^3 x + c \cos x + d</span>:</p><p><span class="math">a = 16</span>, <span class="math">b = -20</span>, <span class="math">c = 5</span>, <span class="math">d = 0</span></p><p>(a) True: <span class="math">a = 16</span></p><p>(b) True: <span class="math">b = -20</span></p><p>(c) True: <span class="math">c = 5</span></p><p>(d) False: <span class="math">d = 0</span> (not 2)</p>
Correct Answer: A, B, C