OCR H240/01 2023 June — Question 7

Exam BoardOCR
ModuleH240/01 (Pure Mathematics)
Year2023
SessionJune
TopicAddition & Double Angle Formulae

7
  1. Use the result \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) to show that \(\cos ( A - B ) = \cos A \cos B + \sin A \sin B\). The function \(\mathrm { f } ( \theta )\) is defined as \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta - 30 ^ { \circ } \right)\), where \(\theta\) is in degrees.
  2. Show that \(f ( \theta ) = \cos ^ { 2 } \theta - \frac { 1 } { 4 }\).
    1. Determine the following.
      • The maximum value of \(\mathrm { f } ( \theta )\)
  3. The smallest positive value of \(\theta\) for which this maximum value occurs
    (ii) Determine the following.
  4. The minimum value of \(\mathrm { f } ( \theta )\)
  5. The smallest positive value of \(\theta\) for which this minimum value occurs