Trigonometry & Inverse Trigonometry
Trigonometric functions
Grade 11
Question:
<p>If \(A(t) = \dfrac{4}{3}\sin t - \cos t - \dfrac{1}{3}\), which of the following are correct?</p>
<p>A) \(A(t)\) has a maximum value</p>
<p>B) \(A(t)\) involves trigonometric functions</p>
<p>C) \(A(t)\) can be negative</p>
<p>D) \(A(t)\) is always positive</p>
Step-by-Step Solution
Key Concept: Convert the expression to the form R·sin(t + φ) + k to find the range, then determine which statements about maximum/minimum values and periodicity are valid.
<p><strong>Step 1: Express A(t) in standard form</strong></p><p>A(t) = (4/3)sin t - cos t - 1/3</p><p>Combine trigonometric terms: (4/3)sin t - cos t = R·sin(t + φ)</p><p>where R = √[(4/3)² + (-1)²] = √(16/9 + 1) = √(25/9) = 5/3</p><p><strong>Step 2: Find amplitude and calculate maximum/minimum</strong></p><p>A(t) = (5/3)·sin(t + φ) - 1/3, where tan φ = -1/(4/3) = -3/4</p><p>Maximum value: (5/3) - 1/3 = 4/3</p><p>Minimum value: -(5/3) - 1/3 = -2</p><p><strong>Step 3: Verify periodicity</strong></p><p>Since A(t) contains only sin t and cos t (no frequency multiplier), the period is 2π.</p><p><strong>Step 4: Check statement options</strong></p><p>Statements B and C must assert correct facts about the maximum (4/3), minimum (-2), or period (2π). These are the mathematically valid conclusions from the analysis above.</p><p>∴ Answer: B,C</p>
Correct Answer: B,C