Trigonometric curve intersections

Questions involving sketching a trigonometric curve (sine or cosine) and another curve (typically linear) to determine number of intersections or solutions.

2 questions

CAIE P1 2023 June Q7
7 A curve has equation \(y = 2 + 3 \sin \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant 4 \pi\).
  1. State greatest and least values of \(y\).
  2. Sketch the curve.
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  3. State the number of solutions of the equation $$2 + 3 \sin \frac { 1 } { 2 } x = 5 - 2 x$$ for \(0 \leqslant x \leqslant 4 \pi\).
OCR C2 Q3
  1. (i) Sketch the curve \(y = \sin x ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
    (ii) Sketch on the same diagram the curve \(y = \sin ( x - 30 ) ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
    (iii) Use your diagram to solve the equation
$$\sin x ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ for \(x\) in the interval \(- 180 \leq x \leq 180\).