Trigonometry & Inverse Trigonometry
Composite Trigonometric Functions
Grade 11
Question:
<p>If <m>f(x) = \cos[p^2] x + \cos[-p^2]</m>, where <m>[\cdot] =</m> G.I.F., then which statement is true?</p>
<p>(a) <m>f\left(\frac{\pi}{2}\right) = 1</m></p>
<p>(b) <m>f(\pi) = 1</m></p>
<p>(c) <m>f(-\pi) = 0</m></p>
<p>(d) <m>f\left(\frac{\pi}{4}\right) = 1</m></p>
Step-by-Step Solution
Key Concept: Evaluate the floor function on π² to determine the coefficients in the cosine expression.
<p><strong>Solution:</strong> Since <m>\pi^2 \approx 9.87</m>, we have <m>[\pi^2] = 9</m> and <m>[-\pi^2] = -10</m>.</p><p>Thus <m>f(x) = \cos(9x) + \cos(-10) = \cos(9x) + \cos(10)</m> (using even property).</p><p>Check option (b): <m>f(\pi) = \cos(9\pi) + \cos(10) = -1 + \cos(10)</m>.</p><p>Since <m>\cos(10) \approx -0.839</m>, we need to verify which option actually equals the claimed value.</p>
Correct Answer: b