Solve trigonometric equation with exact values

A question is this type if and only if it requires solving a trigonometric equation to find exact solutions (involving surds, fractions of π, or exact angle values) in a given interval, possibly using identities or algebraic manipulation.

3 questions · Moderate -0.1

1.05o Trigonometric equations: solve in given intervals
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OCR MEI Paper 3 2022 June Q4
5 marks Standard +0.3
4 In this question you must show detailed reasoning.
Determine the exact solutions of the equation \(2 \cos ^ { 2 } x = 3 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P1 2019 June Q9
10 marks Moderate -0.3
\includegraphics{figure_9} The function f : \(x \mapsto p \sin^2 2x + q\) is defined for \(0 \leqslant x \leqslant \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = \text{f}(x)\).
  1. In terms of \(p\) and \(q\), state the range of f. [2]
  2. State the number of solutions of the following equations.
    1. \(\text{f}(x) = p + q\) [1]
    2. \(\text{f}(x) = q\) [1]
    3. \(\text{f}(x) = \frac{1}{2}p + q\) [1]
  3. For the case where \(p = 3\) and \(q = 2\), solve the equation \(\text{f}(x) = 4\), showing all necessary working. [5]
CAIE P1 2014 November Q11
10 marks Moderate -0.3
The function \(f : x \mapsto 6 - 4\cos(\frac{1}{2}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).
  1. Find the exact value of \(x\) for which \(f(x) = 4\). [3]
  2. State the range of \(f\). [2]
  3. Sketch the graph of \(y = f(x)\). [2]
  4. Find an expression for \(f^{-1}(x)\). [3]